Answer
The solution set is $\{12.15\}$
Work Step by Step
If your calculator doesn't have the $\log_{a} $option,
$\quad $apply change-of-base formula$: \displaystyle \quad \log_{a}M=\frac{\log M}{\log a}$
$\displaystyle \frac{\log(x-1)}{\log 2}-\frac{\log(x+2)}{\log 6}=$2
Graph the function on the LHS.
Graph the function on the RHS.
Find intersections, if there are any.
I graphed with desmos.com/calculator.
Hand calculators have an option such as INTERSECT
to find intersection points.
The solution set is $\{12.15\}$