Example 20. Solve the equation

$$
3^{x-1}+5^{x-1}=34
$$
Solution. Note that $3^{2}+5^{2}=34$. Therefore, equation (20) has a solution $x_{1}=3$. We will prove that there are no other solutions.

Indeed, each of the functions $y=3^{x-1}$ and $y=5^{x-1}$, being exponential functions with a base greater than 1, is increasing; therefore, their sum is also an increasing function. This means that for $x>3$, the value is greater than 34.