Q:

Prove $a^{\log_bc}=c^{\log_ba}$

Can anyone prove
$$a^{\log_bc}=c^{\log_ba}$$
I've tried using algebra but I always get $a=a$ $c=c$ or $b=b$. Sometimes I get the same property but now flipped. Can anyone help?

A:

Take $\log_b$ of both sides:
$$a^{\log_bc}=c^{\log_ba}$$
$$\iff\log_ba^{\log_bc}=\log_bc^{\log_ba}$$
$$\iff\log_bc\log_ba=\log_ba\log_bc$$
This last equality is trivially true by commutativity of multiplication.

A:

Hint:
Take $\log_b$ on both sides.

A:

taking the natural logarithm on both sides we get
$$\log_{b}{c}\ln(a)=\log_{b}{a}\ln(c)$$ and this can be written as
$$\frac{\ln(c)\ln(a)}{\ln(b)}=\frac{\ln(a)\ln(c)}{\ln(b)}$$ which is the same

