# Math Help - logarithmic equation

1. ## logarithmic equation

Show that alogcb =blogca

2. ## Re: logarithmic equation

Originally Posted by Trefoil2727
Show that alogcb =blogca
$\\\text{If }a^{\log_c(b)}=b^{\log_c(a)}\\\text{then }\log_c(b)}\log_c(a)}=\log_c(a)}\log_c(b)}$

3. ## Re: logarithmic equation

A true "direct proof" would go the other way.
From the obviously true $log_c(b)log_c(a)= log_c(a)log_c(b)$, by a "property of logarithms"
$log_c\left(b^{log_c(a)}\right)= log_c\left(a^{log_c(b)}\right)$
and then take c to the power of each side.

Of course, Plato's proof is a perfectly good "synthetic proof" where you start with what you want to prove and manipulate it (here by taking the logarithm, base c, of both sides) to produce something that is "obviously true" and each step is reversible.