# Properties of Logarithms

## Properties of the Logarithmic Function

In this section, we cover many properties of the logarithmic function

 1. ${\displaystyle \log _{a}(1)=0~\log _{a}(a)=1}$
2. ${\displaystyle a^{\log _{a}(M)}=M}$
3. ${\displaystyle \log _{a}(a^{r})=r}$ (Notice this works even for a = 10)
The following properties hold for M, N, and a positive real numbers, ${\displaystyle a\neq 1}$, and r any real number
4. ${\displaystyle \log _{a}(MN)=\log _{a}M+\log _{a}N}$
5. ${\displaystyle \log _{a}\left({\frac {M}{N}}\right)=\log _{a}(M)-\log _{a}(N)}$
6. ${\displaystyle \log _{a}(M^{r})=r\cdot \log _{a}(M)}$
7. ${\displaystyle a^{r}=e^{r\ln(a)}}$
8. If ${\displaystyle M=N,{\text{ then }}\log _{a}(M)=\log _{a}(N)}$
9. If ${\displaystyle \log _{a}(M)=\log _{a}(N),{\text{ then }}M=N}$


## Change of Base Formula

The next two formulas allow us to compare logs of different bases, and are called the change of base formulas.

 If ${\displaystyle a,b\neq 1}$, and M are positive real numbers, then
${\displaystyle \log _{a}(M)={\frac {\log _{b}(M)}{\log _{b}(N)}}}$

 ${\displaystyle \log _{a}(M)={\frac {\log(M)}{\log(a)}}{\text{ and }}\log _{a}(M)={\frac {\ln(M)}{\ln(a)}}}$


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