Rekenregels voor logaritmen
Hier zijn de basis rekenregels die we gaan gebruiken:
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Zorg ervoor dat je deze rekenregels goed begrijpt, want ze zijn onmisbaar bij het herleiden van logaritmische formules.
Voorbeelden van herleidingen
Voorbeeld 1
Herleid de formuletot de vorm
Merk op dat er geen superindex bij de log staat; dit betekent dat het de 10 log is.
Schrijf de formule over:
Zet denaar binnen met de derde rekenregel:
Zet deom in een logaritme: n=\log(t^{2,5})-\log(100000)
Uiteindelijk krijgen we:
Voorbeeld 2
Herleid de formulen=\log_3(81t^5)n=\log(81t^5)tot de vorm
We splitsen de logaritme op: n=\log_3(81)+\log_3(t^5)n=\log_3(81)+\log(t^5)n=\log_3(81)+3\log(t^5)n=\log(81)+3\log(t^5)
Gebruik de derde rekenregel om denaar buiten te halen: n=4+\log_3\left(t^5\right)n=4+\log_3\left(t^5\right)n=4+\log_3\left(t^5\right)n=4+\log_3t^5n=4+\log t^5n=4+\log tn=34+\log tn=3\cdot4+\log tn=3\cdot4+1\log t (Omdat81=3^4)
Dit geeft: n=4+5\log_3\left(t\right)n=4+5\log_3\left(t\right)n=4+5\log_3tn=4+5\log tn=4+15\log tn=+15\log tn=1+15\log t
Voorbeeld 3
Herleid de formulep=0,3+\log_2(6,4q)p=0,3+\log(6,4q)tot
Neem de oorspronkelijke formule over: p=0,3+\log_2(6,4)+\log_2(q)p=0,3+\log_2(6,4)+\log_2(q))p=0,3+(\log_2(6,4)+\log_2(q))p=0,3+(\log_2(6,4)+\log(q))p=0,3+(\log(6,4)+\log(q))
Berekenen reken met de gehele waarde: p=0,3+2,678+\log_2qp=0,3+2,678+\log_2q)p=0,3+(2,678+\log_2q)p=0,3+(2,678+\log q)
Tel nu op: p=2,978+\log_2qp=2,978+\log_2qp=2,978+\log q
Schrijf de\log_2q\log_2\log\log_{2q}\log_2\log^{}'om naar twee 10 logaritmes met behulp van de vierde rekenregel:
p=2{,}978+\frac{\log q}{\log2}p=2{,}978+\frac{\log q}{\log}p=2{,}978+\frac{\log q}{lo}p=2{,}978+\frac{\log q}{l}p=2{,}978+\frac{\log q}{\placeholder{}}p=2{,}978+\frac{\log}{\placeholder{}}p=2{,}978+\frac{\log1}{\placeholder{}}p=2{,}978+\frac{\log}{\placeholder{}}p=2{,}978+\frac{lo}{\placeholder{}}p=2{,}978+\frac{l}{\placeholder{}}p=2{,}978+\frac{\placeholder{}}{\placeholder{}}p=2{,}978+p=2{,}978+p=2{,}978+p=2{,}978+p=2{,}978+p=2{,}978+p=2{,}978p=2{,}97p=2{,}9p=2{,}p=2p=p
p=2{,}978+\frac{1}{\log2}\cdot\log qp=2{,}978+\frac{1}{\log2}\log qp=2{,}978+\frac{1}{\log2}\log qp=2{,}978+\frac{1}{\log2}\log qp=2{,}978+\frac{1}{\log2}\log qp=2{,}978+\frac{1}{\log2}\log qp=2{,}978+\frac{1}{\log2}\log qp=2{,}978+\frac{1}{\log2}+\log qp=2{,}978+\frac{1}{\log2}+\logp=2{,}978+\frac{1}{\log2}+p=2{,}978+\frac{1}{\log2}+p=2{,}978+\frac{1}{\log2}+p=2{,}978+\frac{1}{\log2}+p=2{,}978+\frac{1}{\log2}+p=2{,}978+\frac{1}{\log2}+lp=2{,}978+\frac{1}{\log2}+lop=2{,}978+\frac{1}{\log2}+lp=2{,}978+\frac{1}{\log2}+p=2{,}978+\frac{1}{\log2}p=2{,}978+\frac{1}{\log}p=2{,}978+\frac{1}{\log2}p=2{,}978+\frac{1}{\log}p=2{,}978+\frac{1}{}p=2{,}978+\frac{1}{}p=2{,}978+\frac{1}{}p=2{,}978+\frac{1}{}p=2{,}978+\frac{1}{}p=2{,}978+\frac{1}{}p=2{,}978+\frac{1}{}p=2{,}978+\frac{1}{}p=2{,}978+\frac{1}{}p=2{,}978+\frac{1}{}p=2{,}978+\frac{1}{}p=2{,}978+\frac{1}{\placeholder{}}p=2{,}978+\frac{\placeholder{}}{\placeholder{}}p=2{,}978+p=2{,}978+p=2{,}978+p=2{,}978+p=2{,}978+p=2{,}978+p=2{,}978p=2{,}97p=2{,}9p=2{,}p=2p=ppp
p=2{,}978+3{,}32\cdot\log qp=2{,}978+3{,}32\cdot\logp=2{,}978+3{,}32\cdot\log1p=2{,}978+3{,}32\cdot\logp=2{,}978+3{,}32\cdotp=2{,}978+3{,}32\cdotp=2{,}978+3{,}32\cdotp=2{,}978+3{,}32\cdotp=2{,}978+3{,}32\cdotp=2{,}978+3{,}32p=2{,}978+3{,}32p=2{,}978+3{,}32p=2{,}978+3{,}32p=2{,}978+3{,}32p=2{,}978+3{,}32p=2{,}978+\frac{3{,}32}{}p=2{,}978+\frac{3{,}32}{c}p=2{,}978+\frac{3{,}32}{cd}p=2{,}978+\frac{3{,}32}{c}p=2{,}978+\frac{3{,}32}{\placeholder{}}p=2{,}978+3{,}32p=2{,}978+3{,}3p=2{,}978+3{,}p=2{,}978+3p=2{,}978+p=2{,}978p=2{,}97p=2{,}9p=2{,}p=2p=p
Rekenregels geschikt voor alle rekenvragen
Herleiden van basisformules
•Bereken Beide termen kunnen samengevoegd worden:
•Herleid1,5+\log_9243:1,5+\log243: Zetom in een logaritme: \log_9(9^{1,5})+\log_9(243)\Rightarrow\log_9(27)+\log_9(243)\log_9(9^{1,5})+\log_9(243)\Rightarrow\log_9(27)+\log_9(24)\log_9(9^{1,5})+\log_9(243)\Rightarrow\log_9(27)+\log_9(2)\log_9(9^{1,5})+\log_9(243)\Rightarrow\log_9(27)+\log_9()\log_9(9^{1,5})+\log_9(243)\Rightarrow\log_9(27)+\log_9(3)\log_9(9^{1,5})+\log_9(243)\Rightarrow\log_9(27)+\log_9(3^{})\log_9(9^{1,5})+\log_9(243)\Rightarrow\log_9(27)+\log_9(3^5)\log_9(9^{1,5})+\log_9(243)\Rightarrow\log_9(27)+9\log_9(3^5)\log_9(9^{1,5})+\log_9(243)\Rightarrow\log_9(27)+9\log(3^5)\log_9(9^{1,5})+\log_9(243)\Rightarrow\log_{}(27)+9\log(3^5)\log_9(9^{1,5})+\log_9(243)\Rightarrow\log_{\left(\right)}(27)+9\log(3^5)\log_9(9^{1,5})+\log_9(243)\Rightarrow\log(27)+9\log(3^5)\log(9^{1,5})+\log_9(243)\Rightarrow\log(27)+9\log(3^5)\log(9^{1,5})+\log(243)\Rightarrow\log(27)+9\log(3^5)\log(9^{1,5})+9\log(243)\Rightarrow\log(27)+9\log(3^5)\log(^{1,5})+9\log(243)\Rightarrow\log(27)+9\log(3^5)\log(1^{1,5})+9\log(243)\Rightarrow\log(27)+9\log(3^5) Herleid: \log_9(27\cdot243)=\log_9(6561)=\log_9\left(9^4\right)=4\log_9(27\cdot243)=\log(6561)=\log_9\left(9^4\right)=4\log_9(27\cdot243)=\log(6561)=\log_9\left(9^4=4\right)\log_9(27\cdot243)=\log(6561)=\log_9\left(9=4\right)\log_9(27\cdot243)=\log(6561)=\log_9\left(=4\right)\log_9(27\cdot243)=\log(6561)=\log_9=4\log_9(27\cdot243)=\log(6561)=\log=4\log_9(27\cdot243)=\log(6561)==4\log_9(27\cdot243)=\log(6561)==4\log_9(27\cdot243)=\log(6561)==4\log_9(27\cdot243)=\log(6561)==4\log_9(27\cdot243)=\log(6561)==4\log_9(27\cdot243)=\log(6561)=4
•Vergelijkbare aanpak met\frac{1}{2}\log25+\log_{\frac12}16:\frac{1}{2}\log25+\log_{\frac{1}{\placeholder{}}}16:\frac{1}{2}\log25+\log_116:\frac{1}{2}\log25+\log16:\frac{1}{2}\log25+1\log16:\frac{1}{2}\log25+\frac{1}{}\log16: Zet om naar exponenten: \frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log\left(5\right)-4\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log\left(5\right)-\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log\left(5\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log\left(5\right)+\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log\left(5\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log\left(5\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log\left(\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\rightarrow\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right.\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log(.\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log(2.\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log(20.\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-4}\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)^{-}\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\right)\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\left(\frac12\Rightarrow\log(20)\right)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\frac12\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\frac{1}{\placeholder{}}\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\frac{\placeholder{}}{\placeholder{}}\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(1\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(4\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\left(\Rightarrow\log(20)\right)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac12}\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac{1}{\placeholder{}}}\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{\frac{\placeholder{}}{\placeholder{}}}\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{}\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{}\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{}\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{}\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{}\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{}\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log_{}\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\log\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)+\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(5\right.\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\right.\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\right.\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\sqrt{}\right.\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\sqrt2\right.\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\sqrt{25}\right.\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\sqrt{25}\cdot\right.\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{}\right.\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt1\right.\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right.\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac12}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac{1}{\placeholder{}}}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{\frac{\placeholder{}}{\placeholder{}}}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log_{}(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)\log(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\log(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+1\log(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20)\frac{1}{2}\log(25)+\frac{1}{}\log(16)\Rightarrow\log\left(\sqrt{25}\cdot\sqrt{16}\right)\Rightarrow\log(20) Voeg de logaritmes samen tot één logaritme: \log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=\log\left(0{,}0005^{}\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=\log\left(0{,}000^{}\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=\log\left(0{,}00^{}\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=\log\left(0{,}0^{}\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=\log\left(0{,}^{}\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=\log\left(0^{}\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=\log\left(0.^{}\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=\log\left(0^{}\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=\log\left(^{}\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=\log^{}\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=^{}\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=^{}\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=^{}\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=^{}\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=^{}\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=^{}\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=^{}\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=^{}\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=^{}\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=^{}\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=^{\prime}\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=^{\prime}p\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=^{\prime}\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)=\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10000}\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{1000}\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{100}\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{10}\right)\log(5)-\log\left(10000\right)=\log\left(\frac51\right)\log(5)-\log\left(10000\right)=\log\left(\frac{5}{\placeholder{}}\right)\log(5)-\log\left(10000\right)=\log\left(5\right)\log(5)-\log\left(10000\right)=\log\left(\right)\log(5)-\log\left(10000\right)=\log\log(5)-\log\left(10000\right)=\log(5)-\log\left(10000\right)=\log(5)-\log\left(10000\right)=\log(5)-\log\left(10000\right)=\log(5)-\log\left(10000\right)=\log(5)-\log\left(10000\right)\log(5)-\log\left(10000\right)\log(5)-\log\left(1000\right)\log(5)-\log\left(100\right)\log(5)-\log\left(10\right)\log(5)-\log\left(1\right)\log(5)-\log\left(\right)\log(5)-\log\log(5)-\log(5)-\log(5)-\log(5)-\log(5)-\log(5)
