Question

The sum of the solutions of $3^{x-4}+5^{x-4}=34$ is

Answer 1

Let $f\left(x\right)=3^{x-4}+5^{x-4}-34$
$f^{\prime }\left(x\right)=3^{x-4}\ln 3+5^{x-4}\ln 5>0\forall x\in R$
Thus $f$ is strictly increasing function, so it will have maximum one real root.
It can be observed that $x=6$ is satisfying above equation.
Hence $x=6$ is the only root of the given equation.