Question

Why are the easiest exponential equations to solve those in which both sides are powers of the same base?

Answer 1

Answer:

because it is less work to convert one of the sides to match the others because you can't do anywork without matching bases.

Step-by-step explanation:

Answer 2

Better to see it for yourself :) .

$3^{x-1} = 3^{4 - x}$
can be solved easily by taking the logarithm in base 3 on both sides. We can do that because the log and the exponential functions are strictly increasing. It gives
$x - 1 = 4 - x$
which is easy to solve. 

BUT if you have
$3^{x - 1} = 4^{x - 2}$
then you cannot simply take the logarithm of one base. Let's see why.
$log_{3}(3^{x - 1}) = log_{3}(4^{x - 1})$
gives
$x - 1 = log_{3}(4^{x - 1})$
and the simplification is not easy. I think the easiest way (if not the only) is to change the base of the logarithm with the formula
$log_{a}x = \frac{log_{b}x}{log_{b}a}$

In our example it gives
$x - 1 = \frac{log_{4}(4^{x - 1})}{log_{4}3}$
Which yields
$x - 1 = \frac{x - 1}{log_{4}3}$
From there, it is a simple degree one equation.