Unique solution of algebraic equation

Question

Prove that $10^x+11^x+12^x=13^x+14^x$ has an unique solution over $\mathbb R$. By inspection the equation is true for $x=2$

Answer 1

The equation is equivalent to $$\left(\frac{10}{13}\right)^x+\left(\frac{11}{13}\right)^x+\left(\frac{12}{13}\right)^x=1+\left(\frac{14}{13}\right)^x.$$ It is clear that there are no negative solutions. And for positive $x$, the left side is decreasing, and the right side is increasing, so there is at most one value of $x$ where they are equal.