Finding all solutions of an exponential equation
The easiest way to solve such an equation is taking the logarithm. You will get
$$(x^2-2)\log\left(\left|\frac{x^2-5}{6}\right|\right)=0$$
and the absolute value is needed to avoid the logarithm will take complex values. Then one has to solve
$$x^2-2=0$$
and
$$\frac{x^2-5}{6}=\pm 1.$$
This will provide the full set of solutions.