Find the number of real solutions to the equation $2^x+3^x=5^x$.

Solution

1. Divide the equation $2^x+3^x=5^x$ by $5^x$ to obtain $(2/5{)}^x+(3/5{)}^x=1$.
2. Define the function $f(x)=(2/5{)}^x+(3/5{)}^x$, which is strictly decreasing because both bases are between 0 and 1.
3. Observe that $$\begin{array}{rl}f(1)& =2/5+3/5\\ & =1\end{array}$$, confirming that $x=1$ is a solution.
4. Since $f(x)$ is strictly decreasing, it can take the value 1 at most once.
5. Hence the answer is (B).