Show that a^{logcb }=b^{logca}
A true "direct proof" would go the other way.
From the obviously true $\displaystyle log_c(b)log_c(a)= log_c(a)log_c(b)$, by a "property of logarithms"
$\displaystyle log_c\left(b^{log_c(a)}\right)= log_c\left(a^{log_c(b)}\right)$
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.