Logarithmic properties

Following are the properties of logarithms.

\[\] \(1)~\log_{a}{bc}=\log_{a}{b}+\log_{a}{c}\)

Suppose \(a^x=b\) and \(a^y=c\) such that \[\begin{align}\log_{a}{b}&=x \\ \log_{a}{c}&=y.\end{align}\] Then \(bc=a^x×a^y=a^{x+y}\). Take the logarithm of this expression to get \[\begin{align} \log_{a}{bc}&=x+y \\ &=\log_{a}{b}+\log_{a}{c}. \ _\square \end{align}\]

\[\] \(2)~\log_{a}{\dfrac {b}{c}}=\log_{a}{b}-\log_{a}{c}\)

Suppose \(a^x=b\) and \(a^y=c\) such that \[\begin{align}\log_{a}{b}&=x \\ \log_{a}{c}&=y.\end{align}\] Then \(\dfrac{b}{c}=\dfrac {a^x}{a^y}=a^{x-y}\). Take the logarithm of this expression to get \[\begin{align} \log_{a}{\dfrac {b}{c}}&=x-y\\ &=\log_{a}{b}-\log_{a}{c}. \ _\square \end{align}\]

\[\] \(3)~\log_{a}{b^c}=c\log_{a}{b}\)

Suppose \(\log_{a}{b}=x\) such that \(a^x=b\). Then \(\left(a^x\right)^{c}=b^c \implies a^{xc}=b^{c}\). Take the logarithm of this expression to get \[\begin{align} \log_{a}{b^c}&=xc \\ &=c\log_{a}{b}. \ _\square \end{align}\]

\[\] \(4)~\log_{a}{b^{\frac {1}{r}}}=\dfrac {1}{r}\log_{a}{b}\)

The above can be easily shown by the method in property 3.

\[\] \(5)~\log_{a}{b}=\dfrac {\log_{c}{b}}{\log_{c}{a}}\)

\[\] \(6)~a^{\log_{a}{b}}=b\)