When you check the answers you get from equalities like for example:
$$ ^2\log(x-2) = 3- ^2\log(x)$$
$$ 4^x = 3 \times 2^x + 10$$
so on and so forth, is it sufficient to do the following:
For the $x$'es you get from the logarithmic one, just plug them in $(x-2)$ and $(x)$ and see if it is $>0$.
For the exponential one, check if $x > 0$.
Or do you actually need to plug in your answers in the equality and check if it is correct?