TITLE: Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$
QUESTION [12 upvotes]: Yesterday, my uncle asked me this question:

Prove that $x = 2$ is the unique solution to $3^x + 4^x = 5^x$ where $x \in \mathbb{R}$.

How can we do this? Note that this is not a diophantine equation since $x \in \mathbb{R}$ if you are thinking about Fermat's Last Theorem.

REPLY [4 votes]: One looks for roots of the function $f:x\mapsto a^x+1-b^x$ with $a=\frac34$ and $b=\frac54$.

Since $a\lt1$, the function $x\mapsto a^x$ is decreasing. 
Since $b\gt1$, the function $x\mapsto b^x$ is increasing. 
Hence the function $f$ is decreasing.
And $f(\pm\infty)=\mp\infty$.

As such, the function $f$ has exactly one root. Since $f(0)=1$ this root is positive.