Properties of Logarithms

Properties of the Logarithmic Function

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

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

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

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

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