How can we prove this logarithm property : $\log_{a^m} (b^n) = \frac{n}{m}\log_a b$

Question

How can we prove this logarithm property : $\log_{a^m} (b^n) = \frac{n}{m}\log_a b$

Request you to please guide, will be of great help , not getting any clue how to start this... thanks in advance...

Answer 1

The general technique for working with logarithms is by doing a change-of-base, and make everything into the same base. I will choose base $e$ here and denote the natural logarithm by $\log$. Then, for any $a,b$: $$\log_a(b)=\frac{\log b}{\log a}.$$ We will also use the property that $$\log(a^n)=n\log(a)$$ for all $a,n$. The problem can be done as follows: $$\begin{split}\log_{a^m}(b^n)&=\frac{\log(b^n)}{\log(a^m)}\\&=\frac{n\log b}{m\log a}\\ &=\frac{n}{m}\cdot\frac{\log b}{\log a}\\&=\frac{n}{m}\log_a(b),\end{split}$$ as desired.

Answer 2

Note that $$\begin{align*}(a^m)^{\frac nm \log_a(b)} &= a^{m\cdot\frac nm \log_a(b)} \\ &= a^{ n \log_a(b)} \\ &= (a^{\log_a(b)})^n \\ &= b^n\end{align*}$$

Therefore, by definition, $$\log_{a^m}(b^n) = \frac nm \log_a(b)$$

Answer 3

By definition

$$\log_A B=C \iff A^C=B$$

therefore

$$\log_{a^m} (b^n) = c \iff (a^m)^c=b^n \iff a^{c}=b^{\frac n m}$$

and then using definition again

$$c=\log_a \left(b^{\frac n m}\right)=\frac{n}{m}\log_a b$$