Non-graphical solution to $5\log_{4}a\ + 48\log_{a}4 = \frac{a}{8}$

The logarithms will be rational only if $a$ is of the form $2^b$ where $b$ is an integer

$$f(b)=\dfrac{5b}2+\dfrac{96}b-2^{b-3}$$

Now using AM-GM inequality, the Left hand side $$\ge\sqrt{240}>15$$

$2^{b-3}\ge15\implies b\ge7$

By trial, $b=8$ is a solution