Unknown Exponents

It is easy to check that $x=2$ is a solution. We claim that this is the only solution. The equation is

$10^x+11^x+12^x=13^x+14^x\\$ Divide both sides of the equation by $13^x$ $(\frac{10}{13})^x+ (\frac{11}{13})^x+ (\frac{12}{13})^x=1+(\frac{14}{13})^x$

The left hand side is a decreasing function of $x$ and the right hand side is an increasing function of $x$ .

Therefore their graphs can have at most one point of intersection.