To x

The given equation is similar to: $\left(\frac{5}{12}\right)^{x}+\left(\frac{7}{12}\right)^{x}=1$

$x=1$ is a solution of the equation

$\frac{5}{12}+\frac{7}{12}=1$

Then, for any real number $x>1$ : $\left(\frac{5}{12}\right)^{x}<\frac{5}{12}$ and $\left(\frac{5}{12}\right)^{x}<\frac{7}{12}$ ,

$\left(\frac{5}{12}\right)^{x}+\left(\frac{5}{12}\right)^{x}<\frac{5}{12}+\frac{7}{12}=1$

Therefore, it does not exist a $x>1$ as solution. The same way, it is demonstrated that does not exist solutions for $x<1$ .

Only solution $\boxed{x=1}$

By observation x=1 is a solution.Now to prove that it is the only solution,divide both sides by RHS and we can see an entirely decreasing function is obtained.Thus x=1 is the only real solution