Please find attached problem and my work.
Proof by contradiction. Assume it is rational.
$\displaystyle \log_{8} 6 = \frac{a}{b}$
$\displaystyle 8^{\frac{a}{b}} = 6$
$\displaystyle 8^a = 6^b$
$\displaystyle 2^{3a} = 2^b*3^b$
$\displaystyle 2^{3a-b} = 3^b$
From unique factorization, this is impossible - a contradiction.
QED.