help in finding number of solutions of the equation

Question

I wanted to find the number of solutions of the equation: $$3^{(x-1)} + 5^{(x-1)} = 34$$ I can of course find one solution , but how to be sure that there is just one solution.

Answer 1

The function $f(x)=3^{x-1}+5^{x-1}$ increases monotonically for all $x\in\mathbb{R}$. Hence, if $f(x_0)=34$ for some $x_0\in\mathbb{R}$, then $f(x)<34$ for all $x<x_0$ and $f(x)>34$ for all $x>x_0$.

Answer 2

Hint: It seems to me that one way could be using the Rolle's Theorem for function $$f(x)=3^{x-1}+5^{x-1}-34$$ to get a contradiction.