Finding all real solutions to the equation $3^x+4^x=5^x$ 
Find all real solutions to the equation $$3^x+4^x=5^x.$$

My attempt: It is evident that $x=2$ is a solution.
However, I think that there are no other solutions. So, I define a function $f(x)=3^x+4^x-5^x$.
Differentiating w.r.t $x$, we get $$f'(x)=3^x\ln 3+4^x\ln 4-5^x\ln 5,$$ but that doesn't take me anywhere. Please help. Thank you.  

Hint: Divide through by $5^x$ to produce the equivalent equation
$$\left(\frac{3}{5}\right)^x + \left(\frac{4}{5}\right)^x = 1.$$
The l.h.s. is a strictly decreasing function of $x$.
