jmc

Note that $x=3$ satisfies $3^x+4^x+5^x=6^x.$ We prove that this is the only solution.

Dividing both sides by $6^x,$ we get $$\frac{3^x}{6^x}+\frac{4^x}{6^x}+\frac{5^x}{6^x}=1.$$Let $$f(x)={\left(\frac{3}{6}\right)}^x+{\left(\frac{4}{6}\right)}^x+{\left(\frac{5}{6}\right)}^x.$$Note that the function $f(x)$ is decreasing. We know that $x=\boxed{3}$ is a solution, so it is the unique solution.