Proof of $a^{\log_{a}b}=b$

Question

In the logarithm properties section of the book Calculus for Dummies, there is this property: $$a^{\log_{a}b}=b$$ How can I prove it?

Answer 1

$$b=b$$ $$\implies\log_a b=\log_a b$$ $$\implies(\log_a b)\log_aa=\log_a b$$ As $\log_yy=1$ for any $y$ in the domain $$\implies\log_aa^{\log_ab}=\log_a b$$ It is a rule of logarithms which I hope you know. Now simply if $\log_mn=\log_mp$ It means $n=p$. Using the same reasoning we can say that $$\boxed{a^{\log_{a}b}=b}$$