Schrijf de formuleN=4500\cdot1{,}72^{t+3}N=4500\cdot1{,}72^{t+}N=4500\cdot1{,}72^{t}N=4500\cdot1{,}72^{}N=4500\cdot1{,}72^{\left\lbrace\right\rbrace}N=4500\cdot1{,}72N=4500\cdot1{,}7N=4500\cdot1{,}N=4500\cdot1N=4500\cdotN=4500N=4500N=4500N=4500N=4500N=4500N=450N=45N=4N=Nin de vorm\log_{}\left(N\right)=at+b.\log_{}\left(N\right)=at+b\log_{}\left(N\right)=at+\log_{}\left(N\right)=at\log_{}\left(N\right)=a\log_{}\left(N\right)=\log_{}\left(N\right)\log_{}\left(N\right)=\log_{}\left(N\right)=\log_{}\log_{}\left(N\right)=\log_{}\left(\right)\log_{}\left(N\right)=\log_{}\log_{}\left(N\right)=\log_{\left(\placeholder{}\right)}\log_{}\left(N\right)=\log_{\placeholder{}}\log_{}\left(N\right)=\log_{\placeholder{}}\log_{}\left(N\right)=\log_{\placeholder{}}\left(\right)\log_{}\left(N\right)=lo\left(\right)\log_{}\left(N\right)=l\left(\right)\log_{}\left(N\right)=\left(\right)\log_{}\left(N\right)=\log_{\placeholder{}}\left(\right)\log_{}\left(N\right)=\log_{\placeholder{}}\log_{}\left(N\right)=\log_{}\left(N\right)=\log_{\placeholder{}}\log_{}\left(N\right)=\log_{}\left(N\right)=\log_{\placeholder{}}\log_{}\left(N\right)=lo\log_{}\left(N\right)=l\log_{}\left(N\right)=\log_{}\left(N\right)=-\log_{}\left(N\right)=-\log_{}\left(N\right)\log_{}\left(N\right)\log_{}\left(\right)\log_{}\log_{\left(\placeholder{}\right)}\log_{\placeholder{}}lol\log_{\placeholder{}}lolGeefain vier decimalen enbin twee decimalen.
Leerdoelen
•Je kunt exponentiële formules omwerken.
•Je kunt machtsformules omwerken.
De belangrijke rekenregels voor het omwerken van formules
Voor het omwerken van exponentiële en machtsformules zijn specifieke rekenregels voor logaritmen en machten cruciaal. Een logaritme (\log) is een wiskundige bewerking die de exponent van een grondtal bepaalt die nodig is om een bepaald getal te krijgen. Machten zijn een manier om herhaalde vermenigvuldiging weer te geven.
Rekenregels van logaritmen
•^{g}\log\left(a\right)+^{g}\log\left(b\right)=^{g}\log\left(ab\right)^{g}\log\left(a\right)+^{g}\log\left(b\right)=^{g}\log\left(ab\right)^{g}\log\left(a\right)+^{g}\log\left(b\right)=^{g}\log\left(a\right)^{g}\log\left(a\right)+^{g}\log\left(b\right)=^{g}\log\left(\right)^{g}\log\left(a\right)+^{g}\log\left(b\right)=^{g}\log^{g}\log\left(a\right)+^{g}\log\left(b\right)=^{g}^{g}\log\left(a\right)+^{g}\log\left(b\right)=^{g}\log\left(a\right)+^{g}\log\left(b\right)^{g}\log\left(a\right)+^{g}\log\left(b\right)^{g}\log\left(a\right)+^{g}\log\left(\right)^{g}\log\left(a\right)+^{g}\log^{g}\log\left(a\right)+^{g}^{g}\log\left(a\right)+^{g}\log\left(a\right)^{g}\log\left(a\right)^{g}\log\left(\right)^{g}\log^{g}^{g\log}^{g}
•^{g}\log\left(a\right)-^{g}\log\left(b\right)=^{g}\log\left(\frac{a}{b}\right)^{g}\log\left(a\right)-^{g}\log\left(b\right)=^{g}\log\left(\frac{a}{b}\right)^{g}\log\left(a\right)-^{g}\log\left(b\right)=^{g}\log\left(\frac{a}{\placeholder{}}\right)^{g}\log\left(a\right)-^{g}\log\left(b\right)=^{g}\log\left(a\right)^{g}\log\left(a\right)-^{g}\log\left(b\right)=^{g}\log\left(\right)^{g}\log\left(a\right)-^{g}\log\left(b\right)=^{g}\log^{g}\log\left(a\right)-^{g}\log\left(b\right)=^{g}^{g}\log\left(a\right)-^{g}\log\left(b\right)=^{g}\log\left(a\right)-^{g}\log\left(b\right)^{g}\log\left(a\right)-^{g}\log\left(b\right)^{g}\log\left(a\right)-^{g}\log\left(\right)^{g}\log\left(a\right)-^{g}\log^{g}\log\left(a\right)-^{g}^{g}\log\left(a\right)-^{g}\log\left(a\right)^{g}\log\left(a\right)^{g}\log\left(\right)^{g}\log^{g}^{g\log}^{g}
•p\cdot^{g}\log\left(a\right)=^{g}\log\left(a^{p}\right)p\cdot^{g}\log\left(a\right)=^{g}\log\left(a^{p}\right)p\cdot^{g}\log\left(a\right)=^{g}\log\left(a\right)p\cdot^{g}\log\left(a\right)=^{g}\log\left(\right)p\cdot^{g}\log\left(a\right)=^{g}\logp\cdot^{g}\log\left(a\right)=^{g}p\cdot^{g}\log\left(a\right)=p\cdot^{g}\log\left(a\right)p\cdot^{g}\log\left(a\right)p\cdot^{g}\log\left(\right)p\cdot^{g}\logp\cdot^{g}p\cdot^{g\log}p\cdot^{g}p\cdotp
•^{g}\log\left(a\right)=\frac{^{p}\log\left(a\right)}{^{p}\log\left(g\right)}^{g}\log\left(a\right)=\frac{\left.^{p}\log\left(a\right)\right)}{^{p}\log\left(g\right)}^{g}\log\left(a\right)=\frac{\left(^{p}\log\left(a\right)\right)}{^{p}\log\left(g\right)}^{g}\log\left(a\right)=\frac{\left(^{p}\log\left(a\right)\right)}{^{p}\log\left(g\right)}^{g}\log\left(a\right)=\frac{\left(^{p}\log\left(a\right)\right)}{^{p}\log\left(\right)}^{g}\log\left(a\right)=\frac{\left(^{p}\log\left(a\right)\right)}{^{p}\log\left(b\right)}^{g}\log\left(a\right)=\frac{\left(^{p}\log\left(a\right)\right)}{^{p}\log\left(\right)}^{g}\log\left(a\right)=\frac{\left(^{p}\log\left(a\right)\right)}{^{p}\log}^{g}\log\left(a\right)=\frac{\left(^{p}\log\left(a\right)\right)}{^{p}}^{g}\log\left(a\right)=\frac{\left(^{p}\log\left(a\right)\right)}{}^{g}\log\left(a\right)=\frac{\left(^{p}\log\left(a\right)\right)}{\placeholder{}}^{g}\log\left(a\right)=\frac{\left(^{p}\log\left(a\right)\right)}{\placeholder{}^{}}^{g}\log\left(a\right)=\frac{\left(^{p}\log\left(a\right)\right)}{\placeholder{}^{p}}^{g}\log\left(a\right)=\frac{\left(^{p}\log\left(a\right)\right)}{\placeholder{}}^{g}\log\left(a\right)=\left(^{p}\log\left(a\right)\right)^{g}\log\left(a\right)=\left(^{p}\log\left(a\right)\right.^{g}\log\left(a\right)=\left(^{p}\log\left(a\right)0\right.^{g}\log\left(a\right)=\left(^{p}\log\left(a\right)\right.^{g}\log\left(a\right)=\left(^{p}\log\left(a\right)\right)^{g}\log\left(a\right)=\left(^{p}\log\left(a\right)\right)^{g}\log\left(a\right)=\left(^{p}\log\left(\right)\right)^{g}\log\left(a\right)=\left(^{p}\log\right)^{g}\log\left(a\right)=\left(^{p}\right)^{g}\log\left(a\right)=\left(\right)^{g}\log\left(a\right)=^{g}\log\left(a\right)^{g}\log\left(a\right)^{g}\log\left(\right)^{g}\log^{g}
Rekenregels van machten
•a^{p}\cdot a^{q}=a^{p+q}a^{p}\cdot a^{q}=a^{p+}a^{p}\cdot a^{q}=a^{p}a^{p}\cdot a^{q}=aa^{p}\cdot a^{q}=a^{p}\cdot a^{q}a^{p}\cdot a^{}a^{p}\cdot a^{Q}a^{p}\cdot aa^{p}\cdota^{p}aa\&a\&pa\&aA
•\frac{a^{p}}{a^{q}}=a^{p-q}\frac{a^{p}}{a^{q}}=a^{p-}\frac{a^{p}}{a^{q}}=a^{p}\frac{a^{p}}{a^{q}}=a\frac{a^{p}}{a^{q}}=ap\frac{a^{p}}{a^{q}}=ap-\frac{a^{p}}{a^{q}}=ap\frac{a^{p}}{a^{q}}=a\frac{a^{p}}{a^{q}}=\frac{a^{p}}{a^{q}}\frac{a^{p}}{a}\frac{a^{p}}{\placeholder{}}a^{p}a
•\left(a^{p}\right)^{q}=a^{pq}\left(a^{p}\right)^{q}=a^{p}\left(a^{p}\right)^{q}=a\left(a^{p}\right)^{q}=\left(a^{p}\right)^{q}\left(a^{p}\right)\left(a^{p}\right)\left(a^{p}0\right)\left(a^{p}\right)\left(a\right)\left(\right)
•a^0=1a^0=a^0a
•\frac{1}{a^{p}}=a^{-p}\frac{1}{a^{p}}a^{-p}\frac{1}{a^{p}}a^{-}\frac{1}{a^{p}}a\frac{1}{a^{p}}\frac{1}{a}\frac{1}{\placeholder{}}1
Hoe werk je exponentiële formules om naar de vorm \log\left(N\right)=at+b\log\left(N\right)=at+\log\left(N\right)=at\log\left(N\right)=a\log\left(N\right)=\log\left(N\right)\log\left(N\right)\log\left(\right)\log\left(n\right)\log\left(\right)\log?
Het omwerken van een exponentiële formule naar de vorm \log\left(N\right)=at+b vereist het toepassen van logaritmen aan beide zijden van de formule en vervolgens het gebruik van de rekenregels om de uitdrukking te vereenvoudigen. Hierbij is de afhankelijke variabele, de onafhankelijke variabele en a en b constanten.
Voorbeeld 1a
Gegeven de formule N=2400\cdot1{,}36^{t}N=2400\cdot136^{t}N=2400\cdot1.36^{t}N=24001.36^{t}, moet deze worden omgezet naar .
1.Neem aan beide zijden de logaritme (standaard de 10-log,^{10}\log\left(x\right)^{10}\log\left(x\right)^{10}\log\left(\right)^{10}\log^{10}\log\cdot^{10}\log^{10}^{10\log}^{10}^1^10^1, als er geen grondtal is aangegeven): \log\left(N\right)=\log\left(2400\cdot1{,}36^{t}\right)\log\left(N\right)=\log\left(2400\cdot136^{t}\right)\log\left(N\right)=\log\left(2400\cdot1.36^{t}\right)\log\left(N\right)=\log(2400\cdot1.36^{t}\log\left(N\right)\left.=\log(2400\cdot1.36^{t}\right)\log\left(N\right)\left.=\log(24001.36^{t}\right)\log\left(N\right)\left.=\log(2400*1.36^{t}\right)\log\left(N\right)\left.=(2400*1.36^{t}\right)\log\left(N\right)\left.=l(2400*1.36^{t}\right)\log\left(N\right)\left.=lo(2400*1.36^{t}\right)\log\left(N\right)\left.=log(2400*1.36^{t}\right)\log\left(N\right)\left(=log(2400*1.36^{t}\right)\log N)\left(=log(2400*1.36^{t}\right)\log N\left(=log(2400*1.36^{t}\right)\log\left(=log(2400*1.36^{t}\right)\log=log(2400*1.36^{t})=log(2400*1.36^{t})l=log(2400*1.36^{t})lo=log(2400*1.36^{t})log=log(2400*1.36^{t})
2.Splits de rechterzijde met behulp van de regel ^{g}\log\left(a\right)+^{g}\log\left(b\right)=^{g}\log\left(ab\right): \log\left(N\right)=\log(2400)+\log(1{,}36^{t})\log\left(N\right)=\log(2400)+\log(136^{t})\log\left(N\right)=\log(2400)+\log(1.36^{t})\log\left(N\right)=\log(2400)+(1.36^{t})\log\left(N\right)=\log(2400)+l(1.36^{t})\log\left(N\right)=\log(2400)+lo(1.36^{t})\log\left(N\right)=\log(2400)+log(1.36^{t})\log\left(N\right)=\log(2400+log(1.36^{t})\log\left(N\right)\left.=\log(2400\right)+log(1.36^{t})\log\left(N\right)\left.=(2400\right)+log(1.36^{t})\log\left(N\right)\left.=l(2400\right)+log(1.36^{t})\log\left(N\right)\left.=lo(2400\right)+log(1.36^{t})\log\left(N\right)\left.=log(2400\right)+log(1.36^{t})\log\left(N\right)\left.=log(2400\right)+log(1.36^{t})\log\left(N\right)\left(=log(2400\right)+log(1.36^{t})\log N)\left(=log(2400\right)+log(1.36^{t})\log N\left(=log(2400\right)+log(1.36^{t})\log\left(=log(2400\right)+log(1.36^{t})\log=log(2400)+log(1.36^{t})=log(2400)+log(1.36^{t})l=log(2400)+log(1.36^{t})lo=log(2400)+log(1.36^{t})log=log(2400)+log(1.36^{t})
3.Haal de exponent t voor de logaritme met behulp van de regel p\cdot^{g}\log\left(a\right)=^{g}\log\left(a^{p}\right). De factor komt voor de logaritme: \log\left(N\right)=\log(2400)+t\cdot\log(1{,}36)\log\left(N\right)=\log(2400)+t\cdot\log(136)\log\left(N\right)=\log(2400)+t\cdot\log(1.36)\log\left(N\right)=\log(2400)+t\cdot(1.36)\log\left(N\right)=\log(2400)+t\cdot l(1.36)\log\left(N\right)=\log(2400)+t\cdot lo(1.36)\log\left(N\right)=\log(2400)+t\cdot log(1.36)\log\left(N\right)=\log(2400)+tlog(1.36)\log\left(N\right)=\log(2400)+t*log(1.36)\log\left(N\right)=\log(2400+t*log(1.36)\log\left(N\right)\left.=\log(2400\right)+t*log(1.36)\log\left(N\right)\left.=(2400\right)+t*log(1.36)\log\left(N\right)\left.=l(2400\right)+t*log(1.36)\log\left(N\right)\left.=lo(2400\right)+t*log(1.36)\log\left(N\right)\left.=log(2400\right)+t*log(1.36)\log N)\left.=log(2400\right)+t*log(1.36)\log N)\left.=log(2400\right)+t*log(1.36)\log N)\left(=log(2400\right)+t*log(1.36)\log N\left(=log(2400\right)+t*log(1.36)\log\left(=log(2400\right)+t*log(1.36)\log=log(2400)+t*log(1.36)=log(2400)+t*log(1.36)l=log(2400)+t*log(1.36)lo=log(2400)+t*log(1.36)log=log(2400)+t*log(1.36)
4.Bereken de numerieke waarden. Rond af zoals gevraagd in de opgave: \log(2400)\thickapprox3{,}38\log(2400)\thickapprox338\log(2400)\thickapprox3.38(2400)\thickapprox3.38l(2400)\thickapprox3.38lo(2400)\thickapprox3.38 (afgerond op twee decimalen) \log(1{,}36)\thickapprox0{,}1335\log(1{,}36)\thickapprox01335\log(1{,}36)\thickapprox0.1335\log(136)\thickapprox0.1335\log(1.36)\thickapprox0.1335(1.36)\thickapprox0.1335l(1.36)\thickapprox0.1335lo(1.36)\thickapprox0.1335 (afgerond op vier decimalen)
5.Vul de waarden in en herschik de formule naar de gewenste vorm: \log\left(N\right)=3{,}38+0{,}1335t\log\left(N\right)=3{,}38+01335t\log\left(N\right)=3{,}38+0.1335t\log\left(N\right)=338+0.1335t\log\left(N\right)=3.38+0.1335t\log\left(N=3.38+0.1335t\right)\log\left(=3.38+0.1335t\right)\log=3.38+0.1335t=3.38+0.1335tl=3.38+0.1335tlo=3.38+0.1335tlog=3.38+0.1335t \log\left(N\right)=0{,}1335t+3{,}38\log\left(N\right)=0{,}1335t+338\log\left(N\right)=0{,}1335t+3.38\log\left(N\right)=01335t+3.38\log\left(N\right)=0.1335t+3.38\log\left(N=0.1335t+3.38\right)\log\left(=0.1335t+3.38\right)\log=0.1335t+3.38\log N=0.1335t+3.38N=0.1335t+3.38lN=0.1335t+3.38loN=0.1335t+3.38 Hierbij is a=0{,}1335=0{,}1335A=0{,}1335A=01335 en b=3{,}38b=338b=3.38=3.38.
Voorbeeld 1b
Gegeven de formule N=13000\cdot0{,}56^{\left(t-4\right)}N=13000\cdot0{,}56^{(t-4}N=13000\cdot0{,}56^{(t-}N=13000\cdot0{,}56^{(t}N=13000\cdot0{,}56^{(}N=13000\cdot0{,}56^{(}tN=13000\cdot0{,}56^{(}t-N=13000\cdot0{,}56^{(}t-4N=13000\cdot0{,}56^{(}t-4)N=13000\cdot0{,}6^{(}t-4)N=13000\cdot06^{(}t-4)N=13000\cdot0.6^{(}t-4)N=13000\cdot0.56^{(}t-4)N=130000.56^{(}t-4), moet deze worden omgezet naar .
1.Neem aan beide zijden de logaritme: \log\left(N\right)=\log\left(13000\cdot0{,}56^{\left(t-4\right)}\right)\log\left(N\right)=\log(13000\cdot0{,}56^{\left(t-4\right)}\log\left(N\right)=\log(13000\cdot0{,}56^{\left(t-4\right)}4\log\left(N\right)=\log(13000\cdot0{,}56^{\left(t-4\right)}-4\log\left(N\right)=\log(13000\cdot0{,}56^{\left(t-4\right)}t-4\log\left(N\right)=\log(13000\cdot0{,}56^{(t-4}t-4\log\left(N\right)=\log(13000\cdot0{,}56^{(t-}t-4\log\left(N\right)=\log(13000\cdot0{,}56^{(t}t-4\log\left(N\right)=\log(13000\cdot0{,}56^{(}t-4\log\left(N\right)=\log(13000\cdot0{,}5^{(}t-4\log\left(N\right)=\log(13000\cdot0{,}^{(}t-4\log\left(N\right)=\log(13000\cdot0^{(}t-4\log\left(N\right)=\log(13000\cdot0.^{(}t-4\log\left(N\right)=\log(13000\cdot0.{,}^{(}t-4\log\left(N\right)=\log(13000\cdot0.{,}5^{(}t-4\log\left(N\right)=\log(13000\cdot0.{,}56^{(}t-4\log\left(N\right)=\log(13000\cdot0.{,}5^{(}t-4\log\left(N\right)=\log(13000\cdot0.{,}^{(}t-4\log\left(N\right)=\log(13000\cdot0.^{(}t-4\log\left(N\right)=\log(13000\cdot0.5^{(}t-4\log\left(N\right)=\log(13000\cdot0.56^{(}t-4\log\left(N\right)=\log(130000.56^{(}t-4\log\left(N\right)=\log(13000*0.56^{(}t-4\log\left(N\right)=(13000*0.56^{(}t-4\log\left(N\right)=l(13000*0.56^{(}t-4\log\left(N\right)=lo(13000*0.56^{(}t-4\log\left(N\right)=log(13000*0.56^{(}t-4\log N)=log(13000*0.56^{(}t-4\log\left(N)=log(13000*0.56^{(}t-4\right.\log\left(\left.N)=log(13000*0.56^{(}t-4\right)\right.\log\left(\left.N)=log(13000*0.56^{(}t-4\right)\right)\log\left(\left(N)=log(13000*0.56^{(}t-4\right)\right)\log\left(\left(N=log(13000*0.56^{(}t-4\right)\right)\log N\left(\left.=log(13000*0.56^{(}t-4\right)\right)\log\left(\left.=log(13000*0.56^{(}t-4\right)\right)\log\left.=log(13000*0.56^{(}t-4\right))\log\left.=log(13000*0.56^{(}t-4\right))\log\left.=log(13000*0.56^{(}t-4\right))\log\left(=log(13000*0.56^{(}t-4\right))\log\left(N=log(13000*0.56^{(}t-4\right))\log\left(N)=log(13000*0.56^{(}t-4\right))\log\left(N=log(13000*0.56^{(}t-4\right))\log\left(=log(13000*0.56^{(}t-4\right))\log=log(13000*0.56^{(}t-4))=log(13000*0.56^{(}t-4))l=log(13000*0.56^{(}t-4))lo=log(13000*0.56^{(}t-4))log=log(13000*0.56^{(}t-4))
2.Splits de rechterzijde: \log\left(N\right)=\log(13000)+log\left(0{,}56^{t-4}\right)\log\left(N\right)=\log(13000)+log(0{,}56^{t-4}\log\left(N\right)=\log(13000)+log(0{,}56^{\left(t-4\right.}\log\left(N\right)=\log(13000)+log(0{,}56^{\left(t-4\right)}\log\left(N\right)=\log(13000)+log(0{,}56^{\left(t-4\right)})\log\left(N\right)=\log(13000)+log(0{,}56^{\left(t-4\right)}))\log\left(N\right)=\log(13000)+log(0{,}56^{\left(t-4\right)}4))\log\left(N\right)=\log(13000)+log(0{,}56^{\left(t-4\right)}t4))\log\left(N\right)=\log(13000)+log(0{,}56^{\left(t-4\right)}t-4))\log\left(N\right)=\log(13000)+log(0{,}56^{(t-4}t-4))\log\left(N\right)=\log(13000)+log(0{,}56^{(t-}t-4))\log\left(N\right)=\log(13000)+log(0{,}56^{(t}t-4))\log\left(N\right)=\log(13000)+log(0{,}56^{(}t-4))\log\left(N\right)=\log(13000)+log(056^{(}t-4))\log\left(N\right)=\log(13000)+log(0.56^{(}t-4))\log\left(N\right)=\log(13000+log(0.56^{(}t-4))\log\left(N\right)\left.=\log(13000\right)+log(0.56^{(}t-4))\log\left(N\right)\left.=(13000\right)+log(0.56^{(}t-4))\log\left(N\right)\left.=l(13000\right)+log(0.56^{(}t-4))\log\left(N\right)\left.=lo(13000\right)+log(0.56^{(}t-4))\log\left(N\right)\left.=log(13000\right)+log(0.56^{(}t-4))\log\left(N\right)\left(=log(13000\right)+log(0.56^{(}t-4))\log N)\left(=log(13000\right)+log(0.56^{(}t-4))\log N\left(=log(13000\right)+log(0.56^{(}t-4))\log\left(=log(13000\right)+log(0.56^{(}t-4))\log=log(13000)+log(0.56^{(}t-4))=log(13000)+log(0.56^{(}t-4))l=log(13000)+log(0.56^{(}t-4))lo=log(13000)+log(0.56^{(}t-4))log=log(13000)+log(0.56^{(}t-4))
3.Haal de exponent \left(t-4\right)\left(t-4\right) voor de logaritme. Let op de haakjes, de hele term \left(t-4\right)t-4) is een factor: \log\left(N)=\log\left(13000)+(t-4\right)\cdot\log\left(0{,}56\right)\right.\log\left(N)=\log\left(13000)+(t-4\right)\cdot\log(0{,}56\right.\log\left(N)=\log\left(13000)+(t-4\right)\cdot\log(0{,}56\right)\log\left(N)=\log\left(13000)+(t-4\right)\cdot(0{,}56\right)\log\left(N)=\log\left(13000)+(t-4\right)\cdot l(0{,}56\right)\log\left(N)=\log\left(13000)+(t-4\right)\cdot lo(0{,}56\right)\log\left(N)=\log\left(13000)+(t-4\right)\cdot log(0{,}56\right)\log\left(N)=\log\left(13000)+(t-4\right)\cdot log(056\right)\log\left(N)=\log\left(13000)+(t-4\right)\cdot log(0.56\right)\log\left(N)=\log\left(13000)+(t-4\right)log(0.56\right)\log\left(N)=\log\left(13000)+(t-4\right)*log(0.56\right)\log\left(N)=\log\left(13000+(t-4\right)*log(0.56\right)\log\left(N)=\log\left(1300+(t-4\right)*log(0.56\right)\log\left(N)=\log\left(130+(t-4\right)*log(0.56\right)\log\left(N)=\log\left(13+(t-4\right)*log(0.56\right)\log\left(N)=\log\left(1+(t-4\right)*log(0.56\right)\log\left(N)=\log\left(+(t-4\right)*log(0.56\right)\log\left(N)=\log\left.+(t-4\right)*log(0.56\right)\log\left(N)=\left.+(t-4\right)*log(0.56\right)\log\left(N)\left.+(t-4\right)*log(0.56\right)\log\left.N)\left.+(t-4\right)*log(0.56\right)\log\left.N)\left(+(t-4\right)*log(0.56\right)\log\left.N\left(+(t-4\right)*log(0.56\right)\log\left.\left(+(t-4\right)*log(0.56\right)\log\left.\left.+(t-4\right)*log(0.56\right)\log M\left.\left.+(t-4\right)*log(0.56\right)\log M\left.\left.+(t-4\right)*log(0.56\right)\log M\left.\left.+(t-4\right)*log(0.56\right)\log M\left.\left.+(t-4\right)*log(0.56\right)\log M\left.\left(+(t-4\right)*log(0.56\right)\log M\left.\left(+(t-4\right)*log(0.56\right)\log M\left(\left(+(t-4\right)*log(0.56\right)\log\left(\left(+(t-4\right)*log(0.56\right)\log\left(+(t-4\right)*log(0.56)\log N\left(+(t-4\right)*log(0.56)\log N)\left(+(t-4\right)*log(0.56)\log N\left(+(t-4\right)*log(0.56)\log\left(+(t-4\right)*log(0.56)\log+(t-4)*log(0.56)\log\left(\right.+(t-4)*log(0.56)\log\left(N\right.+(t-4)*log(0.56)\log\left(N)\right.+(t-4)*log(0.56)\log\left(N)=\right.+(t-4)*log(0.56)\log\left(N)=l\right.+(t-4)*log(0.56)\log\left(N)=lo\right.+(t-4)*log(0.56)\log\left(N)=log\right.+(t-4)*log(0.56)\log\left(N)=log(\right.+(t-4)*log(0.56)\log\left(N)=log(1\right.+(t-4)*log(0.56)\log\left(N)=log(13\right.+(t-4)*log(0.56)\log\left(N)=log(130\right.+(t-4)*log(0.56)\log\left(N)=log(1300\right.+(t-4)*log(0.56)\log\left(N)=log(13000\right.+(t-4)*log(0.56)\log\left(N)=log(13000\right)+(t-4)*log(0.56)\log\left(N)log(13000\right)+(t-4)*log(0.56)\log\left(N)-log(13000\right)+(t-4)*log(0.56)\log\left(N)log(13000\right)+(t-4)*log(0.56)\log\left(N)=log(13000\right)+(t-4)*log(0.56)\log\left(N=log(13000\right)+(t-4)*log(0.56)\log\left(=log(13000\right)+(t-4)*log(0.56)\log=log(13000)+(t-4)*log(0.56)=log(13000)+(t-4)*log(0.56)l=log(13000)+(t-4)*log(0.56)lo=log(13000)+(t-4)*log(0.56)log=log(13000)+(t-4)*log(0.56)
4.Bereken \log(13000)(13000)l(13000)lo(13000) en behoud de volledige waarde voor nu (op de rekenmachine): \log\left(N\right)\left.\thickapprox4{,}11394...+(t-4\right)\cdot\log(0{,}56)\log\left(N\right)\left.\thickapprox4{,}11394...+(t-4\right)\cdot\log(056)\log\left(N\right)\left.\thickapprox4{,}11394...+(t-4\right)\cdot\log(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+(t-4\right)\cdot(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+(t-4\right)\cdot l(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+(t-4\right)\cdot lo(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+(t-4\right)\cdot log(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+(t-4\right)log(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+(t-4\right)*log(0.56)\log\left(N\right)\left.\thickapprox411394...+(t-4\right)*log(0.56)\log\left(N\right)\left.\thickapprox4.11394...+(t-4\right)*log(0.56)\log\left(N\right)\left(\thickapprox4.11394...+(t-4\right)*log(0.56)\log N)\left(\thickapprox4.11394...+(t-4\right)*log(0.56)\log N\left(\thickapprox4.11394...+(t-4\right)*log(0.56)\log\left(\thickapprox4.11394...+(t-4\right)*log(0.56)\log\thickapprox4.11394...+(t-4)*log(0.56)\thickapprox4.11394...+(t-4)*log(0.56)l\thickapprox4.11394...+(t-4)*log(0.56)lo\thickapprox4.11394...+(t-4)*log(0.56)log\thickapprox4.11394...+(t-4)*log(0.56)
5.Werk de haakjes uit door \left(t-4\right)t-4) te vermenigvuldigen met \log(0{,}56)\log(056)\log(0.56)(0.56)l(0.56)lo(0.56): \log\left(N\right)\thickapprox4{,}11394...+t\cdot\log\left(0{,}56\right)-4\cdot\log(0{,}56)\log\left(N\right)\thickapprox4{,}11394...+t\cdot\log\left(0{,}56\right)-4\cdot\log(056)\log\left(N\right)\thickapprox4{,}11394...+t\cdot\log\left(0{,}56\right)-4\cdot\log(0.56)\log\left(N\right)\thickapprox4{,}11394...+t\cdot\log\left(0{,}56\right)-4\cdot(0.56)\log\left(N\right)\thickapprox4{,}11394...+t\cdot\log\left(0{,}56\right)-4\cdot l(0.56)\log\left(N\right)\thickapprox4{,}11394...+t\cdot\log\left(0{,}56\right)-4\cdot lo(0.56)\log\left(N\right)\thickapprox4{,}11394...+t\cdot\log\left(0{,}56\right)-4\cdot log(0.56)\log\left(N\right)\thickapprox4{,}11394...+t\cdot\log\left(0{,}56\right)-4log(0.56)\log\left(N\right)\thickapprox4{,}11394...+t\cdot\log\left(0{,}56\right)-4*log(0.56)\log\left(N\right)\thickapprox4{,}11394...+t\cdot\log(0{,}56-4*log(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+t\cdot\log(0{,}56\right)-4*log(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+t\cdot\log(056\right)-4*log(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+t\cdot\log(0.56\right)-4*log(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+t\cdot(0.56\right)-4*log(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+t\cdot l(0.56\right)-4*log(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+t\cdot lo(0.56\right)-4*log(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+t\cdot log(0.56\right)-4*log(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+tlog(0.56\right)-4*log(0.56)\log\left(N\right)\left.\thickapprox4{,}11394...+t*log(0.56\right)-4*log(0.56)\log\left(N\right)\left.\thickapprox411394...+t*log(0.56\right)-4*log(0.56)\log\left(N\right)\left.\thickapprox4.11394...+t*log(0.56\right)-4*log(0.56)\log\left(N\right)\left(\thickapprox4.11394...+t*log(0.56\right)-4*log(0.56)\log N)\left(\thickapprox4.11394...+t*log(0.56\right)-4*log(0.56)\log N\left(\thickapprox4.11394...+t*log(0.56\right)-4*log(0.56)\log\left(\thickapprox4.11394...+t*log(0.56\right)-4*log(0.56)\log\thickapprox4.11394...+t*log(0.56)-4*log(0.56)\thickapprox4.11394...+t*log(0.56)-4*log(0.56)l\thickapprox4.11394...+t*log(0.56)-4*log(0.56)lo\thickapprox4.11394...+t*log(0.56)-4*log(0.56)log\thickapprox4.11394...+t*log(0.56)-4*log(0.56)
6.Bereken t\cdot\log(0{,}56)t\cdot\log(056)t\cdot\log(0.56)t\cdot(0.56)t\cdot l(0.56)t\cdot lo(0.56)t\cdot log(0.56)tlog(0.56) en -4\cdot\log(0{,}56)-4\cdot\log(056)-4\cdot\log(0.56)-4\cdot(0.56)-4\cdot l(0.56)-4\cdot lo(0.56)-4\cdot log(0.56)-4log(0.56) en combineer constante termen. \log(0{,}56)\log(056)\log(0.56)(0.56)l(0.56)lo(0.56) wordt de waarde van A, afgerond op vier decimalen: \log\left(N\right)\left.\thickapprox4{,}11394...+(-0{,}2518\right)t+1.0072...\log\left(N\right)\left.\thickapprox4{,}11394...+(-0{,}2518\right)t+10072...\log\left(N\right)\left.\thickapprox4{,}11394...+(-0{,}2518\right)t+1{,}0072...\log\left(N\right)\left.\thickapprox4{,}11394...+(-0{,}2518\right)t+10072...\log\left(N\right)\left.\thickapprox4{,}11394...+(-0{,}2518\right)t+1.0072...\log\left(N\right)\left.\thickapprox4{,}11394...+(-02518\right)t+1.0072...\log\left(N\right)\left.\thickapprox4{,}11394...+(-0.2518\right)t+1.0072...\log\left(N\right)\left.\thickapprox411394...+(-0.2518\right)t+1.0072...\log\left(N\right)\left.\thickapprox4.11394...+(-0.2518\right)t+1.0072...\log\left(N\right)\left(\thickapprox4.11394...+(-0.2518\right)t+1.0072...\log N)\left(\thickapprox4.11394...+(-0.2518\right)t+1.0072...\log N\left(\thickapprox4.11394...+(-0.2518\right)t+1.0072...\log\left(\thickapprox4.11394...+(-0.2518\right)t+1.0072...\log\thickapprox4.11394...+(-0.2518)t+1.0072...\thickapprox4.11394...+(-0.2518)t+1.0072...l\thickapprox4.11394...+(-0.2518)t+1.0072...lo\thickapprox4.11394...+(-0.2518)t+1.0072...log\thickapprox4.11394...+(-0.2518)t+1.0072...
7.Tel de constante termen bij elkaar op en rond B af op twee decimalen: \log\left(N\right)=-0{,}2518t+5{,}12\log\left(N\right)=-0{,}2518t+512\log\left(N\right)=-0{,}2518t+5.12\log\left(N\right)=-02518t+5.12\log\left(N\right)=-0.2518t+5.12\log\left(N=-0.2518t+5.12\right)\log\left(=-0.2518t+5.12\right)\log=-0.2518t+5.12=-0.2518t+5.12l=-0.2518t+5.12lo=-0.2518t+5.12log=-0.2518t+5.12 Hierbij isa=-0{,}2518a=-02518a=-0.2518=-0.2518 en b=5{,}12b=512b=5.12=5.12.
Alternatieve uitwerking:
1.Herschrijf eerst de macht 0{,}56^{\left(t-4\right)}0{,}56^{\left(t-4\right)}t0{,}56^{\left(t-4\right)}t-0{,}56^{\left(t-4\right)}t-40{,}56^{\left(t-4\right)}t-4)0{,}56^{(t-4}t-4)0{,}56^{(t-}t-4)0{,}56^{(t}t-4)0{,}56^{(}t-4)056^{(}t-4) met de machtsregel a^{p}\cdot a^{q}=a^{p+q} N=13000\cdot0{,}56^{t}\cdot0{,}56^{-4}N=13000\cdot0{,}56^{t}\cdot056^{-4}N=13000\cdot0{,}56^{t}\cdot0.56^{-4}N=13000\cdot056^{t}\cdot0.56^{-4}N=13000\cdot0.56^{t}\cdot0.56^{-4}N=13000\cdot0.56^{t}\cdot0.56^{-}N=13000\cdot0.56^{t}\cdot0.56^{-}4N=13000\cdot0.56^{t}0.56^{-}4N=13000\cdot0.56^{t}*0.56^{-}4N=130000.56^{t}*0.56^{-}4
2.Vermenigvuldig de constante factoren met elkaar: N=(13000\cdot0{,}56^{-4})\cdot0{,}56^{t}N=(13000\cdot056^{-4})\cdot0{,}56^{t}N=(13000\cdot0.56^{-4})\cdot0{,}56^{t}N=(13000\cdot0.56^{-4})\cdot056^{t}N=(13000\cdot0.56^{-4})\cdot0.56^{t}N=(13000\cdot0.56^{-4})0.56^{t}N=(13000\cdot0.56^{-4})*0.56^{t}N=(13000\cdot0.56^{-})*0.56^{t}N=(13000\cdot0.56^{-}4)*0.56^{t}N=(130000.56^{-}4)*0.56^{t} N\thickapprox132193{,}84...\cdot0{,}56^{t}N\thickapprox13219384...\cdot0{,}56^{t}N\thickapprox132193.84...\cdot0{,}56^{t}N\thickapprox132193.84...\cdot056^{t}N\thickapprox132193.84...\cdot0.56^{t}N\thickapprox132193.84...0.56^{t}
3.Neem nu de logaritme aan beide zijden en splits op: \log\left(N\right)=\log\left(132193{,}84...\right)+\log(0{,}56^{t})\log\left(N\right)=\log\left(13219384...\right)+\log(0{,}56^{t})\log\left(N\right)=\log\left(132193.84...\right)+\log(0{,}56^{t})\log\left(N\right)=\log\left(132193.84...\right)+\log(056^{t})\log\left(N\right)=\log\left(132193.84...\right)+\log(0.56^{t})\log\left(N\right)=\log\left(132193.84...\right)+(0.56^{t})\log\left(N\right)=\log\left(132193.84...\right)+l(0.56^{t})\log\left(N\right)=\log\left(132193.84...\right)+lo(0.56^{t})\log\left(N\right)=\log\left(132193.84...\right)+log(0.56^{t})\log\left(N\right)=\log(132193.84...+log(0.56^{t})\log\left(N\right)\left.=\log(132193.84...\right)+log(0.56^{t})\log\left(N\right)\left.=(132193.84...\right)+log(0.56^{t})\log\left(N\right)\left.=l(132193.84...\right)+log(0.56^{t})\log\left(N\right)\left.=lo(132193.84...\right)+log(0.56^{t})\log\left(N\right)\left.=log(132193.84...\right)+log(0.56^{t})\log\left(N\right)\left(=log(132193.84...\right)+log(0.56^{t})\log N)\left(=log(132193.84...\right)+log(0.56^{t})\log N\left(=log(132193.84...\right)+log(0.56^{t})\log\left(=log(132193.84...\right)+log(0.56^{t})\log=log(132193.84...)+log(0.56^{t})=log(132193.84...)+log(0.56^{t})l=log(132193.84...)+log(0.56^{t})lo=log(132193.84...)+log(0.56^{t})log=log(132193.84...)+log(0.56^{t}) \log\left(N\right)=\log\left(132193{,}84...\right)+t\cdot\log\left(0{,}56\right)\log\left(N\right)=\log\left(13219384...\right)+t\cdot\log\left(0{,}56\right)\log\left(N\right)=\log\left(132193.84...\right)+t\cdot\log\left(0{,}56\right)\log\left(N\right)=\log\left(132193.84...\right)+t\cdot\log\left(056\right)\log\left(N\right)=\log\left(132193.84...\right)+t\cdot\log\left(0.56\right)\log\left(N\right)=\log\left(132193.84...\right)+t\cdot\log\left(0.{,}56\right)\log\left(N\right)=\log\left(132193.84...\right)+t\cdot\log\left(0.56\right)\log\left(N\right)=\log\left(132193.84...\right)+t\cdot\log0.56)\log\left(N\right)=\log\left(132193.84...\right)+t\cdot0.56)\log\left(N\right)=\log\left(132193.84...\right)+t\cdot l0.56)\log\left(N\right)=\log\left(132193.84...\right)+t\cdot lo0.56)\log\left(N\right)=\log\left(132193.84...\right)+t\cdot log0.56)\log\left(N\right)=\log\left(132193.84...\right)+t\cdot log(0.56)\log\left(N\right)=\log\left(132193.84...\right)+tlog(0.56)\log\left(N\right)=\log\left(132193.84...\right)+t*log(0.56)\log\left(N\right)=\log(132193.84...+t*log(0.56)\log\left(N\right)\left.=\log(132193.84...\right)+t*log(0.56)\log\left(N\right)\left.=(132193.84...\right)+t*log(0.56)\log\left(N\right)\left.=l(132193.84...\right)+t*log(0.56)\log\left(N\right)\left.=lo(132193.84...\right)+t*log(0.56)\log\left(N\right)\left.=log(132193.84...\right)+t*log(0.56)\log\left(N\right)\left(=log(132193.84...\right)+t*log(0.56)\log N)\left(=log(132193.84...\right)+t*log(0.56)\log N\left(=log(132193.84...\right)+t*log(0.56)\log\left(=log(132193.84...\right)+t*log(0.56)\log=log(132193.84...)+t*log(0.56)=log(132193.84...)+t*log(0.56)l=log(132193.84...)+t*log(0.56)lo=log(132193.84...)+t*log(0.56)log=log(132193.84...)+t*log(0.56)
4.Bereken de waarden en rond af:
\log(132193.84...)\thickapprox5{,}12\log(132193.84...)\thickapprox512\log(132193.84...)\thickapprox5.12(132193.84...)\thickapprox5.12l(132193.84...)\thickapprox5.12lo(132193.84...)\thickapprox5.12 (, afgerond op twee decimalen)
\log(0{,}56)\thickapprox-0{,}2518\log(0{,}56)\thickapprox-02518\log(0{,}56)\thickapprox-0m2518\log(0{,}56)\thickapprox-0m,2518\log(0{,}56)\thickapprox-0m,2518\log(0{,}56)\thickapprox-02518\log(0{,}56)\thickapprox-0.2518\log(056)\thickapprox-0.2518\log(0.56)\thickapprox-0.2518(0.56)\thickapprox-0.2518l(0.56)\thickapprox-0.2518lo(0.56)\thickapprox-0.2518 (, afgerond op vier decimalen)
\log\left(N\right)\left.=5{,}12+(-0{,}2518\right)t
\log\left(N\right)=-0{,}2518t+5{,}12
Voorbeeld 1c
Gegeven de formule \log\left(N\right)=5200\cdot1{,}65^{2t-3}\log\left(N\right)\left.=5200\cdot1{,}65^{2t-3}\right)\log\left(N\right)\left.=5200\cdot1{,}65^{2t-3}-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{2t-3}2-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{2t-3}2t-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{(2t-3}2t-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{(=2t-3}2t-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{(2t-3}2t-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{(2t-}2t-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{(2t}2t-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{(2t-}2t-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{(2t-3}2t-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{(2t-}2t-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{(2t}2t-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{(2}2t-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{(2r}2t-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{(2}2t-\right)\log\left(N\right)\left.=5200\cdot1{,}65^{(}2t-\right)\log\left(N\right)\left.=5200\cdot165^{(}2t-\right)\log\left(N\right)\left.=5200\cdot1.65^{(}2t-\right)\log\left(N\right)\left.=52001.65^{(}2t-\right)\log\left(N\right)\left.=5200*1.65^{(}2t-\right)\log\left(N\right)\left(=5200*1.65^{(}2t-\right)\log N)\left(=5200*1.65^{(}2t-\right)\log N\left(=5200*1.65^{(}2t-\right)\log\left(=5200*1.65^{(}2t-\right)\log=5200*1.65^{(}2t-)=5200*1.65^{(}2t-)3=5200*1.65^{(}2t-), moet deze worden omgezet naar \log\left(N\right)=at+b.
1.Neem aan beide zijden de logaritme: \log\left(N\right)=\log\left(5200\cdot1{,}65^{2t-3}\right)\log\left(N\right)=\log(5200\cdot1{,}65^{2t-3}\log\left(N\right)=\log(5200\cdot1{,}65^{2t-3})\log\left(N\right)=\log(5200\cdot1{,}65^{2t-3}3)\log\left(N\right)\left.=\log(5200\cdot1{,}65^{2t-3}3\right))\log\left(N\right)\left.=\log(5200\cdot1{,}65^{2t-3}-3\right))\log\left(N\right)\left.=\log(5200\cdot1{,}65^{2t-3}t-3\right))\log\left(N\right)\left.=\log(5200\cdot1{,}65^{2t-3}2t-3\right))\log\left(N\right)\left.=\log(5200\cdot1{,}65^{\left(2t-3\right.}2t-3\right))\log\left(N\right)\left.=\log(5200\cdot1{,}65^{\left(2t-3\right)}2t-3\right))\log\left(N\right)\left.=\log(5200\cdot1{,}65^{(2t-3}2t-3\right))\log\left(N\right)\left.=\log(5200\cdot1{,}65^{(2t-}2t-3\right))\log\left(N\right)\left.=\log(5200\cdot1{,}65^{(2t}2t-3\right))\log\left(N\right)\left.=\log(5200\cdot1{,}65^{(2}2t-3\right))\log\left(N\right)\left.=\log(5200\cdot1{,}65^{(}2t-3\right))\log\left(N\right)\left.=\log(5200\cdot165^{(}2t-3\right))\log\left(N\right)\left.=\log(5200\cdot1.65^{(}2t-3\right))\log\left(N\right)\left.=\log(52001.65^{(}2t-3\right))\log\left(N\right)\left.=\log(5200*1.65^{(}2t-3\right))\log\left(N\right)\left.=(5200*1.65^{(}2t-3\right))\log\left(N\right)\left.=l(5200*1.65^{(}2t-3\right))\log\left(N\right)\left.=lo(5200*1.65^{(}2t-3\right))\log\left(N\right)\left.=log(5200*1.65^{(}2t-3\right))\log\left(N\right)\left(=log(5200*1.65^{(}2t-3\right))\log N)\left(=log(5200*1.65^{(}2t-3\right))\log N\left(=log(5200*1.65^{(}2t-3\right))\log\left(=log(5200*1.65^{(}2t-3\right))\log=log(5200*1.65^{(}2t-3))=log(5200*1.65^{(}2t-3))l=log(5200*1.65^{(}2t-3))lo=log(5200*1.65^{(}2t-3))log=log(5200*1.65^{(}2t-3))
2.Splits de rechterzijde: \log\left(N\right)=\log\left(5200\right)+\log(1{,}65^{2t-3})\log\left(N\right)=\log\left(5200\right)+\log(1{,}65^{2t-3}))\log\left(N\right)=\log\left(5200\right)+\log(1{,}65^{2t-3}-))\log\left(N\right)=\log\left(5200\right)+\log(1{,}65^{2t-3}-3))\log\left(N\right)=\log\left(5200\right)+\log(1{,}65^{2t-3}t-3))\log\left(N\right)=\log\left(5200\right)+\log(1{,}65^{2t-3}2t-3))\log\left(N\right)=\log\left(5200\right)+\log(1{,}65^{(2t-3}2t-3))\log\left(N\right)=\log\left(5200\right)+\log(1{,}65^{(2t-}2t-3))\log\left(N\right)=\log\left(5200\right)+\log(1{,}65^{(2t}2t-3))\log\left(N\right)=\log\left(5200\right)+\log(1{,}65^{(2}2t-3))\log\left(N\right)=\log\left(5200\right)+\log(1{,}65^{(}2t-3))\log\left(N\right)=\log\left(5200\right)+\log(165^{(}2t-3))\log\left(N\right)=\log\left(5200\right)+\log(1.65^{(}2t-3))\log\left(N\right)=\log\left(5200\right)+(1.65^{(}2t-3))\log\left(N\right)=\log\left(5200\right)(1.65^{(}2t-3))\log\left(N\right)=\log\left(5200\right)+(1.65^{(}2t-3))\log\left(N\right)=\log\left(5200\right)+l(1.65^{(}2t-3))\log\left(N\right)=\log\left(5200\right)+lo(1.65^{(}2t-3))\log\left(N\right)=\log\left(5200\right)+log(1.65^{(}2t-3))\log\left(N\right)=\log\left(5200\right.+log(1.65^{(}2t-3))\log\left(N\right)=\log\left(5200\right)+log(1.65^{(}2t-3))\log\left(N\right)\left.=\log5200\right)+log(1.65^{(}2t-3))\log\left(N\right)\left.=\log(5200\right)+log(1.65^{(}2t-3))\log\left(N\right)\left.=(5200\right)+log(1.65^{(}2t-3))\log\left(N\right)\left.=l(5200\right)+log(1.65^{(}2t-3))\log\left(N\right)\left.=lo(5200\right)+log(1.65^{(}2t-3))\log\left(N\right)\left.=log(5200\right)+log(1.65^{(}2t-3))\log\left(N\right)\left(=log(5200\right)+log(1.65^{(}2t-3))\log N)\left(=log(5200\right)+log(1.65^{(}2t-3))\log N\left(=log(5200\right)+log(1.65^{(}2t-3))\log\left(=log(5200\right)+log(1.65^{(}2t-3))\log=log(5200)+log(1.65^{(}2t-3))=log(5200)+log(1.65^{(}2t-3))l=log(5200)+log(1.65^{(}2t-3))lo=log(5200)+log(1.65^{(}2t-3))log=log(5200)+log(1.65^{(}2t-3))
3.Haal de exponent () voor de logaritme: \log\left(N)=\log(5200)\right.+(2t-3)\cdot\log(1{,}65)\log\left(N)=\log(5200)\right.+(2t-3)\cdot\log(165)\log\left(N)=\log(5200)\right.+(2t-3)\cdot\log(1.65)\log\left(N)=\log(5200)\right.+(2t-3)\cdot(1.65)\log\left(N)=\log(5200)\right.+(2t-3)\cdot l(1.65)\log\left(N)=\log(5200)\right.+(2t-3)\cdot lo(1.65)\log\left(N)=\log(5200)\right.+(2t-3)\cdot log(1.65)\log\left(N)=\log(5200)\right.+(2t-3)log(1.65)\log\left(N)=\log(5200)\right.+(2t-3)*log(1.65)\log\left(N)=\log(5200\right.+(2t-3)*log(1.65)\log\left(N)=\log(5200\right)+(2t-3)*log(1.65)\log\left(N)=(5200\right)+(2t-3)*log(1.65)\log\left(N)=l(5200\right)+(2t-3)*log(1.65)\log\left(N)=lg(5200\right)+(2t-3)*log(1.65)\log\left(N)=log(5200\right)+(2t-3)*log(1.65)\log\left(N=log(5200\right)+(2t-3)*log(1.65)\log\left(=log(5200\right)+(2t-3)*log(1.65)\log=log(5200)+(2t-3)*log(1.65)=log(5200)+(2t-3)*log(1.65)l=log(5200)+(2t-3)*log(1.65)lo=log(5200)+(2t-3)*log(1.65)log=log(5200)+(2t-3)*log(1.65)
4.Bereken \log(5200)(5200)l(5200)lo(5200) en behoud de volledige waarde: \log\left(N\right)\left.\thickapprox3{,}71600...+(2t-3\right)\cdot\log(1{,}65)\log\left(N\right)\left.\thickapprox371600...+(2t-3\right)\cdot\log(1{,}65)\log\left(N\right)\left.\thickapprox3.71600...+(2t-3\right)\cdot\log(1{,}65)\log\left(N\right)\left.\thickapprox3.71600...+(2t-3\right)\cdot\log(165)\log\left(N\right)\left.\thickapprox3.71600...+(2t-3\right)\cdot\log(1.65)\log\left(N\right)\left.\thickapprox3.71600...+(2t-3\right)\cdot(1.65)\log\left(N\right)\left.\thickapprox3.71600...+(2t-3\right)\cdot l(1.65)\log\left(N\right)\left.\thickapprox3.71600...+(2t-3\right)\cdot lo(1.65)\log\left(N\right)\left.\thickapprox3.71600...+(2t-3\right)\cdot log(1.65)\log\left(N\right)\left.\thickapprox3.71600...+(2t-3\right)log(1.65)\log\left(N\right)\left.\thickapprox3.71600...+(2t-3\right)*log(1.65)\log\left(N\right)\left(\thickapprox3.71600...+(2t-3\right)*log(1.65)\log N)\left(\thickapprox3.71600...+(2t-3\right)*log(1.65)\log N\left(\thickapprox3.71600...+(2t-3\right)*log(1.65)\log\left(\thickapprox3.71600...+(2t-3\right)*log(1.65)\log\thickapprox3.71600...+(2t-3)*log(1.65)\thickapprox3.71600...+(2t-3)*log(1.65)l\thickapprox3.71600...+(2t-3)*log(1.65)lo\thickapprox3.71600...+(2t-3)*log(1.65)log\thickapprox3.71600...+(2t-3)*log(1.65)
5.Werk de haakjes uit: \log\left(N)\thickapprox3{,}71600...+2t\cdot\log\left(1{,}65\right)-3\cdot\log\left(1{,}65\right)\right.\log\left(N)\thickapprox3{,}71600...+2t\cdot\log\left(1{,}65\right)-3\cdot\log(1{,}65\right.\log\left(N)\thickapprox3{,}71600...+2t\cdot\log\left(1{,}65\right)-3\cdot\log(1{,}65\right)\log\left(N)\thickapprox3{,}71600...+2t\cdot\log\left(1{,}65\right)-3\cdot\log(165\right)\log\left(N)\thickapprox3{,}71600...+2t\cdot\log\left(1{,}65\right)-3\cdot\log(1.65\right)\log\left(N)\thickapprox3{,}71600...+2t\cdot\log\left(1{,}65\right)-3\cdot(1.65\right)\log\left(N)\thickapprox3{,}71600...+2t\cdot\log\left(1{,}65\right)-3\cdot l(1.65\right)\log\left(N)\thickapprox3{,}71600...+2t\cdot\log\left(1{,}65\right)-3\cdot lo(1.65\right)\log\left(N)\thickapprox3{,}71600...+2t\cdot\log\left(1{,}65\right)-3\cdot log(1.65\right)\log\left(N)\thickapprox3{,}71600...+2t\cdot\log\left(1{,}65\right)-3log(1.65\right)\log\left(N)\thickapprox3{,}71600...+2t\cdot\log\left(1{,}65\right)-3*log(1.65\right)\log\left(N)\thickapprox3{,}71600...+2t\cdot\log(1{,}65-3*log(1.65\right)\log\left(N)\left.\thickapprox3{,}71600...+2t\cdot\log(1{,}65\right)-3*log(1.65\right)\log\left(N)\left.\thickapprox3{,}71600...+2t\cdot\log(165\right)-3*log(1.65\right)\log\left(N)\left.\thickapprox3{,}71600...+2t\cdot\log(1.65\right)-3*log(1.65\right)\log\left(N)\left.\thickapprox3{,}71600...+2t\cdot(1.65\right)-3*log(1.65\right)\log\left(N)\left.\thickapprox3{,}71600...+2t\cdot l(1.65\right)-3*log(1.65\right)\log\left(N)\left.\thickapprox3{,}71600...+2t\cdot lo(1.65\right)-3*log(1.65\right)\log\left(N)\left.\thickapprox3{,}71600...+2t\cdot log(1.65\right)-3*log(1.65\right)\log\left(N)\left.\thickapprox3{,}71600...+2tlog(1.65\right)-3*log(1.65\right)\log\left(N)\left.\thickapprox3{,}71600...+2t*log(1.65\right)-3*log(1.65\right)\log\left(N)\left.\thickapprox371600...+2t*log(1.65\right)-3*log(1.65\right)\log\left(N)\left.\thickapprox3.71600...+2t*log(1.65\right)-3*log(1.65\right)\log\left(N)\left(\thickapprox3.71600...+2t*log(1.65\right)-3*log(1.65\right)\log\left(N\left(\thickapprox3.71600...+2t*log(1.65\right)-3*log(1.65\right)\log N\left(\thickapprox3.71600...+2t*log(1.65\right)-3*log(1.65)\log\left(\thickapprox3.71600...+2t*log(1.65\right)-3*log(1.65)\log\thickapprox3.71600...+2t*log(1.65)-3*log(1.65)\thickapprox3.71600...+2t*log(1.65)-3*log(1.65)l\thickapprox3.71600...+2t*log(1.65)-3*log(1.65)lo\thickapprox3.71600...+2t*log(1.65)-3*log(1.65)log\thickapprox3.71600...+2t*log(1.65)-3*log(1.65)
6.Bereken de numerieke waarden en rond af waar nodig: 2\cdot\log(1{,}65)\thickapprox2\cdot0{,}21748...\thickapprox0{,}43502\cdot\log(1{,}65)\thickapprox2\cdot0{,}21748...\thickapprox043502\cdot\log(1{,}65)\thickapprox2\cdot0{,}21748...\thickapprox0.43502\cdot\log(1{,}65)\thickapprox2\cdot021748...\thickapprox0.43502\cdot\log(1{,}65)\thickapprox2\cdot0.21748...\thickapprox0.43502\cdot\log(1{,}65)\thickapprox20.21748...\thickapprox0.43502\cdot\log(1{,}65)\thickapprox2*0.21748...\thickapprox0.43502\cdot\log(165)\thickapprox2*0.21748...\thickapprox0.43502\cdot\log(1.65)\thickapprox2*0.21748...\thickapprox0.43502\cdot(1.65)\thickapprox2*0.21748...\thickapprox0.43502\cdot l(1.65)\thickapprox2*0.21748...\thickapprox0.43502\cdot lo(1.65)\thickapprox2*0.21748...\thickapprox0.43502\cdot log(1.65)\thickapprox2*0.21748...\thickapprox0.43502log(1.65)\thickapprox2*0.21748...\thickapprox0.4350 (a, afgerond op vier decimalen)-3\cdot log(1{,}65)\thickapprox-3\cdot0{,}21748...\thickapprox-0{,}65244...-3\cdot log(165)\thickapprox-3\cdot0{,}21748...\thickapprox-0{,}65244...-3\cdot log(1.65)\thickapprox-3\cdot0{,}21748...\thickapprox-0{,}65244...-3log(1.65)\thickapprox-3\cdot0{,}21748...\thickapprox-0{,}65244...-3*log(1.65)\thickapprox-3\cdot0{,}21748...\thickapprox-0{,}65244...-3*log(1.65)\thickapprox-3\cdot0{,}21748...\thickapprox-065244...-3*log(1.65)\thickapprox-3\cdot0{,}21748...\thickapprox-0.65244...-3*log(1.65)\thickapprox-3\cdot0{,}1748...\thickapprox-0.65244...-3*log(1.65)\thickapprox-3\cdot01748...\thickapprox-0.65244...-3*log(1.65)\thickapprox-3\cdot0.1748...\thickapprox-0.65244...-3*log(1.65)\thickapprox-3\cdot0.21748...\thickapprox-0.65244...-3*log(1.65)\thickapprox-30.21748...\thickapprox-0.65244...
7.Combineer de constante termen en rond B af op twee decimalen: \log\left(N\right)\thickapprox3{,}71600...-0{,}65244...+0{,}4350t\log\left(N\right)\thickapprox3{,}71600...-0{,}65244...+04350t\log\left(N\right)\thickapprox3{,}71600...-0{,}65244...+0.4350t\log\left(N\right)\thickapprox3{,}71600...-065244...+0.4350t\log\left(N\right)\thickapprox3{,}71600...-0.65244...+0.4350t\log\left(N\right)\thickapprox371600...-0.65244...+0.4350t\log\left(N\right)\thickapprox3.71600...-0.65244...+0.4350t\log\left(N\thickapprox3.71600...-0.65244...+0.4350t\right)\log\left(\thickapprox3.71600...-0.65244...+0.4350t\right)\log\thickapprox3.71600...-0.65244...+0.4350t\thickapprox3.71600...-0.65244...+0.4350tl\thickapprox3.71600...-0.65244...+0.4350tlo\thickapprox3.71600...-0.65244...+0.4350tlog\thickapprox3.71600...-0.65244...+0.4350t \log\left(N\right)=0{,}4350t+3{,}06\log\left(N\right)=0{,}4350t+306\log\left(N\right)=0{,}4350t+3.06\log\left(N\right)=04350t+3.06\log\left(N\right)=0.4350t+3.06\log\left(N=0.4350t+3.06\right)\log\left(=0.4350t+3.06\right)\log=0.4350t+3.06=0.4350t+3.06l=0.4350t+3.06lo=0.4350t+3.06log=0.4350t+3.06 Hierbij is a=0{,}4350a=04350a=0.4350=0.4350en b=3{,}06=3{,}06B=3{,}06B=306.
Alternatieve uitwerking:
1.Herschrijf eerst de macht 1{,}65^{(2t-3)}165^{(2t-3)}1.65^{(2t-3)}1.65^{(} met machtsregels: N=5200\cdot1.65^{(2t)}\cdot1{,}65^{\left(-3\right)}N=5200\cdot1.65^{(2t)}\cdot1{,}65^{\left(-3\right)}N=5200\cdot1.65^{(2t)}\cdot1{,}65^{\left(-3\right)}-N=5200\cdot1.65^{(2t)}\cdot1{,}65^{\left(-3\right)}-)N=5200\cdot1.65^{(2t)}\cdot1{,}65^{\left(-3\right)}-3)N=5200\cdot1.65^{(2t)}\cdot1{,}65^{(-3}-3)N=5200\cdot1.65^{(2t)}\cdot1{,}65^{(-}-3)N=5200\cdot1.65^{(2t)}\cdot1{,}65^{(}-3)N=5200\cdot1.65^{(2t)}\cdot165^{(}-3)N=5200\cdot1.65^{(2t)}\cdot1.65^{(}-3)N=5200\cdot1.65^{(2t)}1.65^{(}-3)N=5200\cdot1.65^{(2t)}*1.65^{(}-3)N=5200\cdot1.65^{(2t)}2*1.65^{(}-3)N=5200\cdot1.65^{(2t)}2t*1.65^{(}-3)N=5200\cdot1.65^{(2t)}2t)*1.65^{(}-3)N=5200\cdot1.65^{(2t}2t)*1.65^{(}-3)N=5200\cdot1.65^{(2}2t)*1.65^{(}-3)N=5200\cdot1.65^{(}2t)*1.65^{(}-3)N=52001.65^{(}2t)*1.65^{(}-3) N=5200\cdot(1{,}65^2)^{t}\cdot1{,}65^{\left(-3\right)}N=5200\cdot(1{,}65^2)^{t}\cdot1{,}65^{\left(-3\right)}-N=5200\cdot(1{,}65^2)^{t}\cdot1{,}65^{\left(-3\right)}-3N=5200\cdot(1{,}65^2)^{t}\cdot1{,}65^{\left(-3\right)}-3)N=5200\cdot(1{,}65^2)^{t}\cdot1{,}65^{(-3}-3)N=5200\cdot(1{,}65^2)^{t}\cdot1{,}65^{(-}-3)N=5200\cdot(1{,}65^2)^{t}\cdot1{,}65^{(}-3)N=5200\cdot(1{,}65^2)^{t}\cdot165^{(}-3)N=5200\cdot(1{,}65^2)^{t}\cdot1.65^{(}-3)N=5200\cdot(1{,}65^2)^{t}1.65^{(}-3)N=5200\cdot(1{,}65^2)^{t}*1.65^{(}-3)N=5200\cdot(165^2)^{t}*1.65^{(}-3)N=5200\cdot(65^2)^{t}*1.65^{(}-3)N=5200\cdot(.65^2)^{t}*1.65^{(}-3)N=5200\cdot(1.65^2)^{t}*1.65^{(}-3)N=5200(1.65^2)^{t}*1.65^{(}-3) N=(5200\cdot1{,}65^{\left(-3\right)})\cdot(1{,}65^2)^{t}N=(5200\cdot1{,}65^{\left(-3\right)})\cdot(165^2)^{t}N=(5200\cdot1{,}65^{\left(-3\right)})\cdot(1.65^2)^{t}N=(5200\cdot1{,}65^{\left(-3\right)})(1.65^2)^{t}N=(5200\cdot1{,}65^{\left(-3\right)})*(1.65^2)^{t}N=(5200\cdot1{,}65^{\left(-3\right)}-)*(1.65^2)^{t}N=(5200\cdot1{,}65^{\left(-3\right)}-3)*(1.65^2)^{t}N=(5200\cdot1{,}65^{\left(-3\right)}-3))*(1.65^2)^{t}N=(5200\cdot165^{\left(-3\right)}-3))*(1.65^2)^{t}N=(5200\cdot1.65^{\left(-3\right)}-3))*(1.65^2)^{t}N=(5200\cdot1.65^{(-3}-3))*(1.65^2)^{t}N=(5200\cdot1.65^{(-}-3))*(1.65^2)^{t}N=(5200\cdot1.65^{(}-3))*(1.65^2)^{t}N=(52001.65^{(}-3))*(1.65^2)^{t}
2.Bereken de constante factor en het nieuwe grondtal: 5200\cdot1{,}65^{\left(-3\right)}\thickapprox1155{,}97...5200\cdot1{,}65^{\left(-3\right)}\thickapprox115597...5200\cdot1{,}65^{\left(-3\right)}\thickapprox1155.97...5200\cdot1{,}65^{\left(-3\right)}-\thickapprox1155.97...5200\cdot1{,}65^{\left(-3\right)}-3\thickapprox1155.97...5200\cdot1{,}65^{\left(-3\right)}-3)\thickapprox1155.97...5200\cdot1{,}65^{(-3}-3)\thickapprox1155.97...5200\cdot1{,}65^{(-}-3)\thickapprox1155.97...5200\cdot1{,}65^{(}-3)\thickapprox1155.97...5200\cdot165^{(}-3)\thickapprox1155.97...5200\cdot1.65^{(}-3)\thickapprox1155.97...52001.65^{(}-3)\thickapprox1155.97... 1{,}65^2=2{,}72251{,}65^2=272251{,}65^2=2.7225165^2=2.7225 N\thickapprox1155{,}97...\cdot2{,}7225^{t}N\thickapprox1155{,}97...\cdot27225^{t}N\thickapprox1155{,}97...\cdot2.7225^{t}N\thickapprox1155{,}97...2.7225^{t}N\thickapprox1155{,}97...*2.7225^{t}N\thickapprox115597...*2.7225^{t}
3.Neem nu de logaritme aan beide zijden: \log\left(N)=\log\left(1155{,}97...\right)\right.+\log(2{,}7225^{t})\log\left(N)=\log\left(1155{,}97...\right)\right.+\log(27225^{t})\log\left(N)=\log\left(1155{,}97...\right)\right.+\log(2.7225^{t})\log\left(N)=\log\left(1155{,}97...\right)\right.+(2.7225^{t})\log\left(N)=\log\left(1155{,}97...\right)\right.+l(2.7225^{t})\log\left(N)=\log\left(1155{,}97...\right)\right.+lo(2.7225^{t})\log\left(N)=\log\left(1155{,}97...\right)\right.+log(2.7225^{t})\log\left(N)=\log(1155{,}97...\right.+log(2.7225^{t})\log\left(N)=\log(1155{,}97...\right)+log(2.7225^{t})\log\left(N)=\log(115597...\right)+log(2.7225^{t})\log\left(N)=\log(1155.97...\right)+log(2.7225^{t})\log\left(N)=(1155.97...\right)+log(2.7225^{t})\log\left(N)=l(1155.97...\right)+log(2.7225^{t})\log\left(N)=lo(1155.97...\right)+log(2.7225^{t})\log\left(N)=log(1155.97...\right)+log(2.7225^{t})\log\left(N=log(1155.97...\right)+log(2.7225^{t})\log\left(=log(1155.97...\right)+log(2.7225^{t})\log=log(1155.97...)+log(2.7225^{t})=log(1155.97...)+log(2.7225^{t})l=log(1155.97...)+log(2.7225^{t})lo=log(1155.97...)+log(2.7225^{t})log=log(1155.97...)+log(2.7225^{t}) \log\left(N)=\log\left(1155{,}97...\right)\right.+t\cdot\log(2{,}7225)\log\left(N)=\log\left(1155{,}97...\right)\right.+t\cdot\log(27225)\log\left(N)=\log\left(1155{,}97...\right)\right.+t\cdot\log(2.7225)\log\left(N)=\log\left(1155{,}97...\right)\right.+t\cdot(2.7225)\log\left(N)=\log\left(1155{,}97...\right)\right.+t\cdot o(2.7225)\log\left(N)=\log\left(1155{,}97...\right)\right.+t\cdot og(2.7225)\log\left(N)=\log\left(1155{,}97...\right)\right.+tog(2.7225)\log\left(N)=\log\left(1155{,}97...\right)\right.+t*og(2.7225)\log\left(N)=\log\left(1155{,}97...\right)\right.+t*log(2.7225)\log\left(N)=\log(1155{,}97...\right.+t*log(2.7225)\log\left(N)=\log(1155{,}97...\right)+t*log(2.7225)\log\left(N)=\log(115597...\right)+t*log(2.7225)\log\left(N)=\log(1155.97...\right)+t*log(2.7225)\log\left(N)=(1155.97...\right)+t*log(2.7225)\log\left(N)=l(1155.97...\right)+t*log(2.7225)\log\left(N)=lo(1155.97...\right)+t*log(2.7225)\log\left(N)=log(1155.97...\right)+t*log(2.7225)\log\left(N=log(1155.97...\right)+t*log(2.7225)\log\left(=log(1155.97...\right)+t*log(2.7225)\log=log(1155.97...)+t*log(2.7225)=log(1155.97...)+t*log(2.7225)l=log(1155.97...)+t*log(2.7225)lo=log(1155.97...)+t*log(2.7225)log=log(1155.97...)+t*log(2.7225)
4.Bereken de waarden en rond af:
\log(1155{,}97...)\thickapprox3{,}06\log(1155{,}97...)\thickapprox306\log(1155{,}97...)\thickapprox3.06\log(115597...)\thickapprox3.06\log(1155.97...)\thickapprox3.06(1155.97...)\thickapprox3.06l(1155.97...)\thickapprox3.06lo(1155.97...)\thickapprox3.06 (bn, afgerond op twee decimalen)
\log(2{,}7225)\thickapprox0{,}4350\log(2{,}7225)\thickapprox04350\log(2{,}7225)\thickapprox0.4350\log(27225)\thickapprox0.4350\log(2.7225)\thickapprox0.4350(2.7225)\thickapprox0.4350l(2.7225)\thickapprox0.4350lo(2.7225)\thickapprox0.4350 (a, afgerond op vier decimalen) \log\left(N\right)=0{,}4350t+3{,}06\log\left(N\right)=0{,}4350t+306\log\left(N\right)=0{,}4350t+3.06\log\left(N\right)=04350t+3.06\log\left(N\right)=0.4350t+3.06\log\left(N=0.4350t+3.06\right)\log\left(=0.4350t+3.06\right)\log=0.4350t+3.06=0.4350t+3.06l=0.4350t+3.06lo=0.4350t+3.06log=0.4350t+3.06
Hoe werk je machtsformules om naar de vorm\log\left(y\right)=a+b\cdot\log\left(x\right)\log\left(y\right)=a+b\cdot\log\left(x\right)X\log\left(y\right)=a+b\cdot\log\left(xX\right)\log\left(y\right)=a+b\cdot\log\left(X\right)\log\left(y\right)=a+b\cdot\log X\log\left(y\right)=a+b\cdot X\log\left(y\right)=a+b\cdot lX\log\left(y\right)=a+b\cdot loX\log\left(y\right)=a+b\cdot logX\log\left(y\right)=a+blogX\log\left(y\right)=a+b*logX\log\left(y\right)=a+*logX\log\left(y\right)=a+B*logX\log\left(y\right)=aq+B*logX\log\left(y\right)=aq+B*logX\log\left(y\right)=+B*logX\log\left(y\right)=A+B*logX\log\left(y=A+B*logX\right)\log\left(y\left.\right)=A+B*logX\right)\log\left(y\left.\right)=A+B*logX\right)\log\left(y\left(\right)=A+B*logX\right)\log\left(y\left(=A+B*logX\right)\right)\log\left(y=A+B*logX\right)\log\left(=A+B*logX\right)\log\left(Y=A+B*logX\right)\log\left(=A+B*logX\right)\log=A+B*logX=A+B*logX?
Voor het omwerken van een machtsformuley=cx^{n}y=cxy=cy=y=yy-y77=7==7==y7==7=7naar de vorm\log\left(y\right)=a+b\cdot\log\left(x\right), neem je de logaritme van beide zijden en pas je de rekenregels toe. Hierbij is y de afhankelijke variabele, x de onafhankelijke variabele en a en b constanten.
Voorbeeld 2a
Gegeven de formuley=20x^{1{,}6}y=20x^{1{,}6}.y=20x^{1{,}6}.6y=20x^{1{,}}.6y=20x^1.6y=20x^1.6y=20^1.6y=20X^1.6y=20*X^1.6=20*X^1.6, moet deze worden omgezet naar \log\left(y\right)=a+b\cdot\log\left(x\right).
1.Neem aan beide zijden de logaritme: \log\left(y\right)=\log\left(20x^{1{,}6}\right)\log\left(y\right)=\log(20x^{1{,}6}\log\left(y\right)=\log(20x^{1{,}6}.\log\left(y\right)=\log(20x^{1{,}6}.6\log\left(y\right)\left.=\log(20x^{1{,}6}.6\right)\log\left(y\right)\left.=\log(20x^{1{,}}.6\right)\log\left(y\right)\left.=\log(20x^1.6\right)\log\left(y\right)\left.=\log(20^1.6\right)\log\left(y\right)\left.=\log(20X^1.6\right)\log\left(y\right)\left.=\log(20*X^1.6\right)\log\left(y\right)\left.=(20*X^1.6\right)\log\left(y\right)\left.=l(20*X^1.6\right)\log\left(y\right)\left.=lo(20*X^1.6\right)\log\left(y\right)\left.=log(20*X^1.6\right)\log\left(y\right)\left(=log(20*X^1.6\right)\log y)\left(=log(20*X^1.6\right)\log y\left(=log(20*X^1.6\right)\log\left(=log(20*X^1.6\right)\log=log(20*X^1.6)=log(20*X^1.6)l=log(20*X^1.6)lo=log(20*X^1.6)log=log(20*X^1.6)
2.Splits de rechterzijde: \log\left(y\right)=\log(20)+\log\left(x^{1{,}6})\right.\log\left(y\right)=\log(20)+\log\left(x^{1{,}6})X\right.\log\left(y\right)=\log(20)+\log\left(x^{1{,}6})X^{}\right.\log\left(y\right)=\log(20)+\log\left(x^{1{,}6})X^{}\right)\log\left(y\right)=\log(20)+\log\left(x^{1{,}6})X^1\right)\log\left(y\right)=\log(20)+\log\left(x^{1{,}6})X^1.\right)\log\left(y\right)=\log(20)+\log\left(x^{1{,}6})X^1.6\right)\log\left(y\right)=\log(20)+\log\left(x^{1{,}6}X^1.6\right)\log\left(y\right)=\log(20)+\log\left(x_{}^{1{,}6}X^1.6\right)\log\left(y\right)=\log(20)+\log\left(x_{)}^{1{,}6}X^1.6\right)\log\left(y\right)=\log(20)+\log\left(x^{1{,}6}X^1.6\right)\log\left(y\right)=\log(20)+\log\left(x^{1{,}6}(X^1.6\right)\log\left(y\right)=\log(20)+\log\left(x^{1{,}6}X^1.6\right)\log\left(y\right)=\log(20)+\log\left(x^{1{,}6)}X^1.6\right)\log\left(y\right)=\log(20)+\log\left(x^{1{,}6}X^1.6\right)\log\left(y\right)=\log(20)+\log\left(x^{1{,}}X^1.6\right)\log\left(y\right)=\log(20)+\log\left(x^1X^1.6\right)\log\left(y\right)=\log(20)+\log\left(x^1,X^1.6\right)\log\left(y\right)=\log(20)+\log\left(x^1X^1.6\right)\log\left(y\right)=\log(20)+\log\left(xX^1.6\right)\log\left(y\right)=\log(20)+\log\left(X^1.6\right)\log x\left(y\right)=\log(20)+\log\left(X^1.6\right)\log\left(y\right)=\log(20)+\log\left(X^1.6\right)\log\left(y\right)=\log(20)+\log X^1.6)\log\left(y\right)=\log(20)+X^1.6)\log\left(y\right)=\log(20)+lX^1.6)\log\left(y\right)=\log(20)+loX^1.6)\log\left(y\right)=\log(20)+logX^1.6)\log\left(y\right)=\log(20)+log(X^1.6)\log\left(y\right)=\log(20+log(X^1.6)\log\left(y\right)\left.=\log(20\right)+log(X^1.6)\log\left(y\right)\left.=(20\right)+log(X^1.6)\log\left(y\right)\left.=l(20\right)+log(X^1.6)\log\left(y\right)\left.=lo(20\right)+log(X^1.6)\log\left(y\right)\left.=log(20\right)+log(X^1.6)\log\left(y\right)\left(=log(20\right)+log(X^1.6)\log y)\left(=log(20\right)+log(X^1.6)\log y\left(=log(20\right)+log(X^1.6)\log\left(=log(20\right)+log(X^1.6)\log=log(20)+log(X^1.6)=log(20)+log(X^1.6)l=log(20)+log(X^1.6)lo=log(20)+log(X^1.6)log=log(20)+log(X^1.6) Belangrijk: De exponent 1,6 staat alleen bij , niet bij 20. Splits daarom eerst en haal de exponent daarna pas naar voren.
3.Haal de exponent 1,6 voor de logaritme: \log\left(y)=\log(20)+1{,}6\log\left(x\right)\right.\log\left(y)=\log(20)+1{,}6\log(x\right.\log\left(y)=\log(20)+1{,}6\log(x\right)\log\left(y)=\log(20)+1{,}6\log(\right)\log\left(y)=\log(20)+1{,}6\log(X\right)\log\left(y)=\log(20)+1{,}6(X\right)\log\left(y)=\log(20)+1{,}6\cdot(X\right)\log\left(y)=\log(20)+1{,}6\cdot l(X\right)\log\left(y)=\log(20)+1{,}6\cdot lo(X\right)\log\left(y)=\log(20)+1{,}6\cdot log(X\right)\log\left(y)=\log(20)+1{,}6log(X\right)\log\left(y)=\log(20)+1{,}6*log(X\right)\log\left(y)=\log(20)+16*log(X\right)\log\left(y)=\log(20)+1.6*log(X\right)\log\left(y)=\log(20+1.6*log(X\right)\log\left(y)=\left.\log(20\right)+1.6*log(X\right)\log\left(y)=\left.(20\right)+1.6*log(X\right)\log\left(y)=\left.l(20\right)+1.6*log(X\right)\log\left(y)=\left.lo(20\right)+1.6*log(X\right)\log\left(y)=\left.log(20\right)+1.6*log(X\right)\log\left(y)\left.log(20\right)+1.6*log(X\right)\log\left(y\left.log(20\right)+1.6*log(X\right)\log\left(\left.log(20\right)+1.6*log(X\right)\log\left(y\left.log(20\right)+1.6*log(X\right)\log\left(y\left.log(20\right)+1.6*log(X\right)\log\left(y\left.log(20\right)+1.6*log(X\right)\log\left(y\left.log(20\right)+1.6*log(X\right)\log\left(y\left.=log(20\right)+1.6*log(X\right)\log\left(y\left.=log(20\right)+1.6*log(X\right)\log\left(y\left.=log(20\right)+1.6*log(X\right)\log\left(y\left.=log(20\right)+1.6*log(X\right)\log\left(y\left.=log(20\right)+1.6*log(X\right)\log\left(y\left.=log(20\right)+1.6*log(X\right)\log\left(y\left.)=log(20\right)+1.6*log(X\right)\log\left(y\left.)=log(20\right)+1.6*log(X\right)\log\left(y\left.)=log(20\right)+1.6*log(X\right)\log y\left.)=log(20\right)+1.6*log(X)\log y\left.=log(20\right)+1.6*log(X)\log y\left(=log(20\right)+1.6*log(X)\log\left(=log(20\right)+1.6*log(X)\log=log(20)+1.6*log(X)\log9=log(20)+1.6*log(X)\log=log(20)+1.6*log(X)=log(20)+1.6*log(X)l=log(20)+1.6*log(X)lo=log(20)+1.6*log(X)log=log(20)+1.6*log(X)
4.Bereken \log(20)(20)l(20)lo(20) en rond af op twee decimalen: \log(20)\thickapprox1{,}30\log(20)\thickapprox130\log(20)\thickapprox1.30(20)\thickapprox1.30l(20)\thickapprox1.30lo(20)\thickapprox1.30(dit wordt a)
5.Vul de waarden in: \log\left(y\right)=1{,}30+1{,}6\cdot\log\left(x\right)\log\left(y\right)=1{,}30+1{,}6\cdot\log\left(x\right)\log\left(y\right)=1{,}30+1{,}6\cdot\log\left(\right)\log\left(y\right)=1{,}30+1{,}6\cdot\log\log\left(y\right)=1{,}30+1{,}6\cdot\log\left(y\right)=1{,}30+1{,}6\cdot X\log\left(y\right)=1{,}30+1{,}6\cdot lX\log\left(y\right)=1{,}30+1{,}6\cdot loX\log\left(y\right)=1{,}30+1{,}6\cdot logX\log\left(y\right)=1{,}30+1{,}6logX\log\left(y\right)=1{,}30+1{,}6*logX\log\left(y\right)=1{,}30+16*logX\log\left(y\right)=1{,}30+1.6*logX\log\left(y\right)=130+1.6*logX\log\left(y\right)=1.30+1.6*logX\log\left(y=1.30+1.6*logX\right)\log\left(=1.30+1.6*logX\right)\log=1.30+1.6*logX\log Y=1.30+1.6*logXY=1.30+1.6*logXlY=1.30+1.6*logXloY=1.30+1.6*logX Hierbij is a=1{,}30a=130a=1.30=1.30 en b=1{,}6=1{,}6B=1{,}6B=16.
Voorbeeld 2b
Gegeven de formule y=78\cdot x^{-3{,}9}y=78\cdot x^{-3{,}9}3y=78\cdot x^{-3{,}9}3.y=78\cdot x^{-3{,}9}3.9y=78\cdot x^{-3{,}}3.9y=78\cdot x^{-3}3.9y=78\cdot x^{-}3.9y=78\cdot^{-}3.9y=78\cdot X^{-}3.9y=78X^{-}3.9y=78*X^{-}3.9=78*X^{-}3.9, moet deze worden omgezet naar \log\left(y\right)=a+b\cdot\log\left(x\right).
1.Neem aan beide zijden de logaritme: \log\left(y)=\log\left(78\cdot x^{-3{,}9}\right)\right.\log\left(y)=\log(78\cdot x^{-3{,}9}\right.\log\left(y)=\log(78\cdot x^{-3{,}9}\right)\log\left(y)=\log(78\cdot x^{-3{,}9}3\right)\log\left(y)=\log(78\cdot x^{-3{,}9}3.\right)\log\left(y)=\log(78\cdot x^{-3{,}9}3.9\right)\log\left(y)=\log(78\cdot x^{-3{,}}3.9\right)\log\left(y)=\log(78\cdot x^{-3}3.9\right)\log\left(y)=\log(78\cdot x^{-}3.9\right)\log\left(y)=\log(78\cdot^{-}3.9\right)\log\left(y)=\log(78\cdot X^{-}3.9\right)\log\left(y)=\log(78X^{-}3.9\right)\log\left(y)=\log(78*X^{-}3.9\right)\log\left(y)=(78*X^{-}3.9\right)\log\left(y)=l(78*X^{-}3.9\right)\log\left(y)=lo(78*X^{-}3.9\right)\log\left(y)=log(78*X^{-}3.9\right)\log\left(y=log(78*X^{-}3.9\right)\log\left(=log(78*X^{-}3.9\right)\log=log(78*X^{-}3.9)=log(78*X^{-}3.9)=log(78*X^{-}3.9)\log=log(78*X^{-}3.9)l=log(78*X^{-}3.9)lo=log(78*X^{-}3.9)log=log(78*X^{-}3.9)
2.Splits de rechterzijde: \log\left(y)=\log(78)+\log\left(x^{-3{,}9}\right)\right.\log\left(y)=\log(78)+\log\left(x^{-3{,}9}\right)3\right.\log\left(y)=\log(78)+\log\left(x^{-3{,}9}\right)3\right)\log\left(y)=\log(78)+\log\left(x^{-3{,}9}\right)3.\right)\log\left(y)=\log(78)+\log\left(x^{-3{,}9}\right)3.9\right)\log\left(y)=\log(78)+\log\left(x^{-3{,}9}3.9\right)\right)\log\left(y)=\log(78)+\log\left(x^{-3{,}}3.9\right)\right)\log\left(y)=\log(78)+\log\left(x^{-3}3.9\right)\right)\log\left(y)=\log(78)+\log\left(x^{-}3.9\right)\right)\log\left(y)=\log(78)+\log\left(^{-}3.9\right)\right)\log\left(y)=\log(78)+\log^{-}3.9\right)\log\left(y)=\log(78)+\log X^{-}3.9\right)\log\left(y)=\log(78)+\log(X^{-}3.9\right)\log\left(y)=\log(78)+(X^{-}3.9\right)\log\left(y)=\log(78)+l(X^{-}3.9\right)\log\left(y)=\log(78)+lo(X^{-}3.9\right)\log\left(y)=\log(78)+log(X^{-}3.9\right)\log\left(y)=\log(78+log(X^{-}3.9\right)\log\left(y)\left.=\log(78\right)+log(X^{-}3.9\right)\log\left(y)\left.=(78\right)+log(X^{-}3.9\right)\log\left(y)\left.=l(78\right)+log(X^{-}3.9\right)\log\left(y)\left.=lo(78\right)+log(X^{-}3.9\right)\log\left(y)\left.=log(78\right)+log(X^{-}3.9\right)\log\left(y)\left(=log(78\right)+log(X^{-}3.9\right)\log\left(y\left(=log(78\right)+log(X^{-}3.9\right)\log y\left(=log(78\right)+log(X^{-}3.9)\log\left(=log(78\right)+log(X^{-}3.9)\log=log(78)+log(X^{-}3.9)=log(78)+log(X^{-}3.9)l=log(78)+log(X^{-}3.9)lo=log(78)+log(X^{-}3.9)log=log(78)+log(X^{-}3.9)
3.Haal de exponent -3,9 voor de logaritme: \log\left(y\right)=\log(78)+(-3{,}9)\cdot\log(x)\log\left(y\right)=\log(78)+(-3{,}9)\cdot\log()\log\left(y\right)=\log(78)+(-3{,}9)\cdot\log(X)\log\left(y\right)=\log(78)+(-3{,}9)\cdot(X)\log\left(y\right)=\log(78)+(-3{,}9)\cdot l(X)\log\left(y\right)=\log(78)+(-3{,}9)\cdot lo(X)\log\left(y\right)=\log(78)+(-3{,}9)\cdot log(X)\log\left(y\right)=\log(78)+(-3{,}9)log(X)\log\left(y\right)=\log(78)+(-3{,}9)*log(X)\log\left(y\right)=\log(78)+(-39)*log(X)\log\left(y\right)=\log(78)+(-3.9)*log(X)\log\left(y\right)=\log(78+(-3.9)*log(X)\log\left(y\right)\left.=\log(78\right)+(-3.9)*log(X)\log\left(y\right)\left.=(78\right)+(-3.9)*log(X)\log\left(y\right)\left.=(78\right)+(-3.9)*log(X)\log\left(y\right)\left.=lo(78\right)+(-3.9)*log(X)\log\left(y\right)\left.=log(78\right)+(-3.9)*log(X)\log\left(y\right)\left(=log(78\right)+(-3.9)*log(X)\log y)\left(=log(78\right)+(-3.9)*log(X)\log y\left(=log(78\right)+(-3.9)*log(X)\log\left(=log(78\right)+(-3.9)*log(X)\log=log(78)+(-3.9)*log(X)=log(78)+(-3.9)*log(X)l=log(78)+(-3.9)*log(X)lo=log(78)+(-3.9)*log(X)log=log(78)+(-3.9)*log(X) \log\left(y\right)=\log(78)-3{,}9\cdot\log(x)\log\left(y\right)=\log(78)-3{,}9\cdot\log()\log\left(y\right)=\log(78)-3{,}9\cdot\log(X)\log\left(y\right)=\log(78)-3{,}9\cdot(X)\log\left(y\right)=\log(78)-3{,}9\cdot l(X)\log\left(y\right)=\log(78)-3{,}9\cdot lo(X)\log\left(y\right)=\log(78)-3{,}9\cdot log(X)\log\left(y\right)=\log(78)-3{,}9log(X)\log\left(y\right)=\log(78)-3{,}9*log(X)\log\left(y\right)=\log(78)-39*log(X)\log\left(y\right)=\log(78)-9*log(X)\log\left(y\right)=\log(78)-.9*log(X)\log\left(y\right)=\log(78)-3.9*log(X)\log\left(y\right)=\log(78-3.9*log(X)\log\left(y\right)\left.=\log(78\right)-3.9*log(X)\log\left(y\right)\left.=(78\right)-3.9*log(X)\log\left(y\right)\left.=l(78\right)-3.9*log(X)\log\left(y\right)\left.=lo(78\right)-3.9*log(X)\log\left(y\right)\left.=log(78\right)-3.9*log(X)\log y)\left.=log(78\right)-3.9*log(X)\log y)\left(=log(78\right)-3.9*log(X)\log y\left(=log(78\right)-3.9*log(X)\log\left(=log(78\right)-3.9*log(X)\log=log(78)-3.9*log(X)=log(78)-3.9*log(X)l=log(78)-3.9*log(X)lo=log(78)-3.9*log(X)log=log(78)-3.9*log(X)
4.Bereken \log(78)(78)l(78)lo(78) en rond af op twee decimalen: \log(78)\thickapprox1{,}89\log(78)\thickapprox189\log(78)\thickapprox1.89(78)\thickapprox1.89l(78)\thickapprox1.89lo(78)\thickapprox1.89 (dit wordt aA)
5.Vul de waarden in: \log\left(y\right)=1{,}89-3{,}9\cdot\log\left(x\right)\log\left(y\right)=1{,}89-3{,}9\cdot\log\left(x\right)\log\left(y\right)=1{,}89-3{,}9\cdot\log\left(\right)\log\left(y\right)=1{,}89-3{,}9\cdot\log\log\left(y\right)=1{,}89-3{,}9\cdot\log X\log\left(y\right)=1{,}89-3{,}9\cdot X\log\left(y\right)=1{,}89-3{,}9\cdot lX\log\left(y\right)=1{,}89-3{,}9\cdot loX\log\left(y\right)=1{,}89-3{,}9\cdot logX\log\left(y\right)=1{,}89-3{,}9logX\log\left(y\right)=1{,}89-3{,}9*logX\log\left(y\right)=1{,}89-39*logX\log\left(y\right)=1{,}89-3.9*logX\log\left(y\right)=189-3.9*logX\log\left(y\right)=1.89-3.9*logX\log\left(y=1.89-3.9*logX\right)\log\left(=1.89-3.9*logX\right)\log=1.89-3.9*logX=1.89-3.9*logXl=1.89-3.9*logXlo=1.89-3.9*logXlog=1.89-3.9*logX Hierbij is a=1{,}89a=1{,}8a=1{,}a=1a=a en b=-3{,}9b=-39b=-3.9=-3.9.
Voorbeeld 2c
Gegeven de formule y=\frac{320}{x^3\sqrt{x}}y=\frac{320}{x^3\sqrt{\placeholder{}}}y=\frac{320}{x^3}y=\frac{320}{x^{}}y=\frac{320}{x^2}y=\frac{320}{x^{23}}y=\frac{320}{x^2}y=\frac{320}{x}y=\frac{320}{\placeholder{}}y=320y=32y=3y=y, moet deze worden omgezet naar \log\left(y\right)=a+b\cdot\log\left(x\right).
1.Herschrijf de formule eerst naar de vorm y=cx^{n}. Dit betekent dat de breuk moet worden vereenvoudigd en de noemer als een negatieve exponent moet worden geschreven.
\sqrt{x}\sqrt{\placeholder{}}WWoWorWortWorteWortel is hetzelfde als x^{\frac12}x^{\frac{1}{\placeholder{}}}x^1x^{1.}x^1xof x^{0{,}5}x^{0{,}}x^0x^0,x^0x.
Combineer de machten van x in de noemer: x^3\cdot x^{0{,}5}=x^{\left(3+0{,}5\right)}=x^{3{,}5}x^3\cdot x^{0{,}5}=x^{\left(3+0{,}5\right)}=^{3{,}5}x^3\cdot x^{0{,}5}=x^{\left(3+0{,}5\right)}=X^{3{,}5}x^3\cdot x^{0{,}5}=x^{\left(3+0{,}5\right)}=X^{3{,}5}.x^3\cdot x^{0{,}5}=x^{\left(3+0{,}5\right)}=X^{3{,}5}.5x^3\cdot x^{0{,}5}=x^{\left(3+0{,}5\right)}=X^{3{,}}.5x^3\cdot x^{0{,}5}=x^{\left(3+0{,}5\right)}=X^3.5x^3\cdot x^{0{,}5}=x^{\left(3+0{,}5\right)}3=X^3.5x^3\cdot x^{0{,}5}=x^{\left(3+0{,}5\right)}3)=X^3.5x^3\cdot x^{0{,}5}=x^{\left(3+0{,}5\right)}3+)=X^3.5x^3\cdot x^{0{,}5}=x^{\left(3+0{,}5\right)}3+5)=X^3.5x^3\cdot x^{0{,}5}=x^{\left(3+0{,}5\right)}3+05)=X^3.5x^3\cdot x^{0{,}5}=x^{\left(3+0{,}5\right)}3+0.5)=X^3.5x^3\cdot x^{0{,}5}=x^{(3+0{,}5}3+0.5)=X^3.5x^3\cdot x^{0{,}5}=x^{(3+0{,}}3+0.5)=X^3.5x^3\cdot x^{0{,}5}=x^{(3+0}3+0.5)=X^3.5x^3\cdot x^{0{,}5}=x^{(3+}3+0.5)=X^3.5x^3\cdot x^{0{,}5}=x^{(3}3+0.5)=X^3.5x^3\cdot x^{0{,}5}=x^{(}3+0.5)=X^3.5x^3\cdot x^{0{,}5}=^{(}3+0.5)=X^3.5x^3\cdot x^{0{,}5}=X^{(}3+0.5)=X^3.5x^3\cdot x^{0{,}5}.=X^{(}3+0.5)=X^3.5x^3\cdot x^{0{,}5}.5=X^{(}3+0.5)=X^3.5x^3\cdot x^{0{,}}.5=X^{(}3+0.5)=X^3.5x^3\cdot x^0.5=X^{(}3+0.5)=X^3.5x^3\cdot^0.5=X^{(}3+0.5)=X^3.5x^3\cdot X^0.5=X^{(}3+0.5)=X^3.5x^3X^0.5=X^{(}3+0.5)=X^3.5x^3*X^0.5=X^{(}3+0.5)=X^3.5^3*X^0.5=X^{(}3+0.5)=X^3.5 (met de regel a^{p}\cdot a^{q}=a^{p+q}).
De formule wordt y=\frac{320}{x^{3{,}5}}=\frac{320}{x^{3{,}5}}7=\frac{320}{x^{3{,}5}}=\frac{320}{x^{3{,}5}}Y=\frac{320}{x^{3{,}5}}Y=\frac{320}{x^{3{,}5}}XY=\frac{320}{x^{3{,}5}}X^{}Y=\frac{320}{x^{3{,}5}}X^{}.Y=\frac{320}{x^{3{,}5}}X^{}.5Y=\frac{320}{x^{3{,}5}}X^3.5Y=\frac{320}{x^{3{,}}}X^3.5Y=\frac{320}{x^3}X^3.5Y=\frac{320}{x}X^3.5Y=\frac{320}{\placeholder{}}X^3.5Y=320X^3.5
1.Haal de uit de noemer door de exponent van teken te veranderen (met de regel \frac{1}{a^{p}}=a^{-p}): y=320\cdot x^{-3{,}5}y=320\cdot x^{-3{,}5}3y=320\cdot x^{-3{,}5}3.y=320\cdot x^{-3{,}5}3.5y=320\cdot x^{-3{,}}3.5y=320\cdot x^{-3}3.5y=320\cdot x^{-}3.5y=320\cdot^{-}3.5y=320\cdot X^{-}3.5y=320X^{-}3.5y=320*X^{-}3.5=320*X^{-}3.5
2.Neem aan beide zijden de logaritme: \log\left(y)=\log\left(320\cdot x^{-3{,}5}\right)\right.\log\left(y)=\log(320\cdot x^{-3{,}5}\right.\log\left(y)=\log(320\cdot x^{-3{,}5}\right)\log\left(y)=\log(320\cdot x^{-3{,}5}3\right)\log\left(y)=\log(320\cdot x^{-3{,}5}3.\right)\log\left(y)=\log(320\cdot x^{-3{,}5}3.5\right)\log\left(y)=\log(320\cdot x^{-3{,}}3.5\right)\log\left(y)=\log(320\cdot x^{-3}3.5\right)\log\left(y)=\log(320\cdot x^{-}3.5\right)\log\left(y)=\log(320\cdot^{-}3.5\right)\log\left(y)=\log(320\cdot X^{-}3.5\right)\log\left(y)=\log(320X^{-}3.5\right)\log\left(y)=\log(320*X^{-}3.5\right)\log\left(y)=(320*X^{-}3.5\right)\log\left(y)=\sin(320*X^{-}3.5\right)\log\left(y)=(320*X^{-}3.5\right)\log\left(y)=l(320*X^{-}3.5\right)\log\left(y)=lo(320*X^{-}3.5\right)\log\left(y)=log(320*X^{-}3.5\right)\log\left(y=log(320*X^{-}3.5\right)\log\left(=log(320*X^{-}3.5\right)\log=log(320*X^{-}3.5)=log(320*X^{-}3.5)l=log(320*X^{-}3.5)lo=log(320*X^{-}3.5)log=log(320*X^{-}3.5)
3.Splits de rechterzijde: \log\left(y\right)=\log(320)+\log\left(x^{-3{,}5}\right)\log\left(y\right)=\log(320)+\log\left(x^{-3{,}5}\right)3\log\left(y\right)=\log(320)+\log\left(x^{-3{,}5}\right)3.\log\left(y\right)=\log(320)+\log\left(x^{-3{,}5}\right)3.5\log\left(y\right)=\log(320)+\log\left(x^{-3{,}5}\right)3.5)\log\left(y\right)=\log(320)+\log(x^{-3{,}5}3.5)\log\left(y\right)=\log(320)+\log(x^{-3{,}}3.5)\log\left(y\right)=\log(320)+\log(x^{-3}3.5)\log\left(y\right)=\log(320)+\log(x^{-}3.5)\log\left(y\right)=\log(320)+\log(^{-}3.5)\log\left(y\right)=\log(320)+\log(X^{-}3.5)\log\left(y\right)=\log(320)+(X^{-}3.5)\log\left(y\right)=\log(320)+l(X^{-}3.5)\log\left(y\right)=\log(320)+lo(X^{-}3.5)\log\left(y\right)=\log(320)+log(X^{-}3.5)\log\left(y\right)=\log(320+log(X^{-}3.5)\log\left(y\right)\left.=\log(320\right)+log(X^{-}3.5)\log\left(y\right)\left.=(320\right)+log(X^{-}3.5)\log\left(y\right)\left.=l(320\right)+log(X^{-}3.5)\log\left(y\right)\left.=lo(320\right)+log(X^{-}3.5)\log\left(y\right)\left.=log(320\right)+log(X^{-}3.5)\log\left(y\right)\left(=log(320\right)+log(X^{-}3.5)\log y)\left(=log(320\right)+log(X^{-}3.5)\log y\left(=log(320\right)+log(X^{-}3.5)\log\left(=log(320\right)+log(X^{-}3.5)\log=log(320)+log(X^{-}3.5)=log(320)+log(X^{-}3.5)l=log(320)+log(X^{-}3.5)lo=log(320)+log(X^{-}3.5)log=log(320)+log(X^{-}3.5)
4.Haal de exponent -3,5 voor de logaritme: \log\left(y\right)=\log(320)+(-3{,}5)\cdot\log(x)\log\left(y\right)=\log(320)+(-3{,}5)\cdot\log()\log\left(y\right)=\log(320)+(-3{,}5)\cdot\log(X)\log\left(y\right)=\log(320)+(-3{,}5)\cdot(X)\log\left(y\right)=\log(320)+(-3{,}5)\cdot l(X)\log\left(y\right)=\log(320)+(-3{,}5)\cdot lo(X)\log\left(y\right)=\log(320)+(-3{,}5)\cdot log(X)\log\left(y\right)=\log(320)+(-3{,}5)log(X)\log\left(y\right)=\log(320)+(-3{,}5)*log(X)\log\left(y\right)=\log(320)+(-35)*log(X)\log\left(y\right)=\log(320)+(-3.5)*log(X)\log\left(y\right)=\log(320+(-3.5)*log(X)\log\left(y\right)\left.=\log(320\right)+(-3.5)*log(X)\log\left(y\right)\left.=(320\right)+(-3.5)*log(X)\log\left(y\right)\left.=l(320\right)+(-3.5)*log(X)\log\left(y\right)\left.=lo(320\right)+(-3.5)*log(X)\log\left(y\right)\left.=log(320\right)+(-3.5)*log(X)\log\left(y\right)\left(=log(320\right)+(-3.5)*log(X)\log y)\left(=log(320\right)+(-3.5)*log(X)\log y\left(=log(320\right)+(-3.5)*log(X)\log\left(=log(320\right)+(-3.5)*log(X)\log=log(320)+(-3.5)*log(X)=log(320)+(-3.5)*log(X)l=log(320)+(-3.5)*log(X)lo=log(320)+(-3.5)*log(X)log=log(320)+(-3.5)*log(X) \log\left(y\right)=\log(320)-3{,}5\cdot\log(x)\log\left(y\right)=\log(320)-3{,}5\cdot\log()\log\left(y\right)=\log(320)-3{,}5\cdot\log(X)\log\left(y\right)=\log(320)-3{,}5\cdot(X)\log\left(y\right)=\log(320)-3{,}5\cdot l(X)\log\left(y\right)=\log(320)-3{,}5\cdot lo(X)\log\left(y\right)=\log(320)-3{,}5\cdot log(X)\log\left(y\right)=\log(320)-3{,}5log(X)\log\left(y\right)=\log(320)-3{,}5*log(X)\log\left(y\right)=\log(320)-35*log(X)\log\left(y\right)=\log(320)-3.5*log(X)\log\left(y\right)=\log(320-3.5*log(X)\log\left(y\right)\left.=\log(320\right)-3.5*log(X)\log\left(y\right)\left.=(320\right)-3.5*log(X)\log\left(y\right)\left.=l(320\right)-3.5*log(X)\log\left(y\right)\left.=lo(320\right)-3.5*log(X)\log\left(y\right)\left.=log(320\right)-3.5*log(X)\log\left(y\right)\left(=log(320\right)-3.5*log(X)\log y)\left(=log(320\right)-3.5*log(X)\log y\left(=log(320\right)-3.5*log(X)\log\left(=log(320\right)-3.5*log(X)\log=log(320)-3.5*log(X)=log(320)-3.5*log(X)l=log(320)-3.5*log(X)lo=log(320)-3.5*log(X)log=log(320)-3.5*log(X)
5.Bereken \log(320)(320)l(320)lo(320) en rond af op twee decimalen: \log(320)\thickapprox2{,}51\log(320)\thickapprox251\log(320)\thickapprox2.51(320)\thickapprox2.51l(320)\thickapprox2.51lo(320)\thickapprox2.51 (dit wordt a)
6.Vul de waarden in: \log\left(y\right)=2{,}51-3{,}5\cdot\log\left(x\right)\log\left(y\right)=2{,}51-3{,}5\cdot\log\left(x\right)X\log\left(y\right)=2{,}51-3{,}5\cdot\log\left(xX\right)\log\left(y\right)=2{,}51-3{,}5\cdot\log\left(X\right)\log\left(y\right)=2{,}51-3{,}5\cdot\log X\log\left(y\right)=2{,}51-3{,}5\cdot X\log\left(y\right)=2{,}51-3{,}5\cdot lX\log\left(y\right)=2{,}51-3{,}5\cdot loX\log\left(y\right)=2{,}51-3{,}5\cdot logX\log\left(y\right)=2{,}51-3{,}5logX\log\left(y\right)=2{,}51-3{,}5*logX\log\left(y\right)=2{,}51-35*logX\log\left(y\right)=2{,}51-3.5*logX\log\left(y\right)=251-3.5*logX\log\left(y\right)=2.51-3.5*logX\log\left(y=2.51-3.5*logX\right)\log\left(=2.51-3.5*logX\right)\log=2.51-3.5*logX=2.51-3.5*logXl=2.51-3.5*logXlo=2.51-3.5*logXlog=2.51-3.5*logX Hierbij is a=2{,}51a=251a=2.51=2.51 en b=-3{,}5b=-35b=-3.5=-3.5.
