Graphing would have been my method of choice, as well.
Interesting side note, while I believe $x=4$ is the only actual solution, if you account for complex numbers, I think there are an infinite number of values of $x$ that come arbitrarily close to solving the equation.
Let $f(x) = 5^{\sqrt{x}}+12^{\sqrt{x}}-13^{\sqrt{x}}$
If we let $t \equiv \sqrt{-x}$, then
$f(x) = g(t) = 5^{it} + 12^{it} - 13^{it} = e^{\ln(5) it} + e^{\ln(12) it} - e^{\ln(13) it}$,
or the sum of three sinusoidal waves at frequencies (in t) of ln(5), ln(12), and ln(13).
Since these frequencies are irrational, I don't think the phases will ever exactly synch up except at $x=t=0$, so the magnitude will never be identically zero. It will come arbitrarily close over and over again, though.
For example, $|f(-28854)| = 0.0149$. At more negative numbers, the local minima start getting smaller. Unfortunately, the "beat" frequency is so low that you have to go to really large negative values of $x$ to find significantly smaller magnitudes of f. They are more or less predictable, however, so we could find them were we so inclined.