How to prove that $a^{\log_cb}=b^{\log_ca}$

Question

I've met a question whereby it asked me to show that $a^{\log_cb}=b^{\log_ca}$. I'm okay with the other logarithm questions. But I don't know how to show this question out. Can anyone give some hints or explanation for me? Thanks in advance.

Answer 1

Note that for any base $c$ $$a^b = c^{b\log_c a}$$ You can use this here to see that both are equal to $$c^{\log_c a \log_c b}$$

Answer 2

This follows from $c^{\log_c a\log_c b} = c^{\log_c b\log_c a}$.

Answer 3

Take the $\log_c$ of both sides: $$ \log_c(a^{\log_cb})\qquad\log_c(b^{\log_ca}) $$ Use the property of the logarithm: $$ \log_cb\log_ca\qquad\log_ca\log_cb $$ These are equal, so also the original terms are.