Solve exponential equation $3^{x-1}+5^{x-1}=34$

Question

What should I do? If we divide say with $3^{x-1}$, we win nothing, considering the $34$. How do we solve this equation?

Thanks.

Answer 1

It is clear at first sight that $x=3$ is a solution so the only one by the increasing function $3^x+5^x$. Trying to find a "deduction" we see first at the integers and from $$3^x+(3+2)^x=3^x+3^x +K$$ where K is positive, one has $2<x<4$ because $3^4+3^4>34$ hence the end (although it had been more interesting that x be an irrational).

Answer 2

Hint: $3^{x-1}+5^{x-1}=3^2+5^2 $ also, think about the graph of $\alpha ^x$, what you can dedude?