Find the number of natural solutions of $5^x+7^x+11^x=6^x+8^x+9^x$ 
Find the number of natural solutions of $5^x+7^x+11^x=6^x+8^x+9^x$

It's easy to see that $x=0$ and $x=1$ are solutions but are these the only one? How do I demonstrate that?
I've tried to write them either:
$$5^x+7^x+11^x=2^x*3^x+2^{3x}+3^{2x}$$
or
$$5^x+7^x+11^x=(5+1)^x+(7+1)^x+(11-2)^x$$
and tried to think of some AM-GM mean inequality or to divide everything by $11^x$, but those don't seem like the way to go. Any hints?

Let $f(x)=x^k,$ where $k>1$ or $k<0$.
Thus, $f$ is a convex function and since $(11,7,5)\succ(9,8,6),$ by Karamata we obtain:
$$f(11)+f(7)+f(5)>f(9)+f(8)+f(6).$$
Also, for $0<k<1$ we see that $f$ is a concave function.
Thus, by Karamata again
$$f(11)+f(7)+f(5)<f(9)+f(8)+f(6).$$
Thus, it remains to check, what happens for $k\in\{0,1\}$.
