Show that $a^{\log_c b}= b^{\log_c a}$

Question

Show that $a^{\log_c b}= b^{\log_c a}$.

I start from LHS and add $\log a$ on it, but it leave $\log_c b$. Then I have not idea about how to continue it(maybe my working is wrong...

Can anybody solve this question?

Answer 1

Let $\displaystyle\log_cb=B\implies c^B=b$

and $\displaystyle\log_ca=A\implies c^A=a$

$\displaystyle\implies b^{\log_ca}=b^A=(c^B)^A=(c^A)^B=a^{\log_cb}$

Answer 2

Hint: $\log_cb=\dfrac{\ln b}{\ln c}$ and $a^x=e^{x\ln a}$