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 $$\begin{aligned}\log_{a}{b}&=x \\ \log_{a}{c}&=y.\end{aligned}$$ Then $bc=a^x×a^y=a^{x+y}$. Take the logarithm of this expression to get $$\begin{aligned} \log_{a}{bc}&=x+y \\ &=\log_{a}{b}+\log_{a}{c}. \ _\square \end{aligned}$$

$$$$ $2)~\log_{a}{\dfrac {b}{c}}=\log_{a}{b}-\log_{a}{c}$

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

$$$$ $3)~\log_{a}{b^c}=c\log_{a}{b}$

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 $$\begin{aligned} \log_{a}{b^c}&=xc \\ &=c\log_{a}{b}. \ _\square \end{aligned}$$

$$$$ $4)~\log_{a}{b^{\frac {1}{r}}}=\dfrac {1}{r}\log_{a}{b}$

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

$$$$ $5)~\log_{a}{b}=\dfrac {\log_{c}{b}}{\log_{c}{a}}$

$$$$ $6)~a^{\log_{a}{b}}=b$