Solve the equation $2^{\mathrm{x}}+3^{\mathrm{x}}=5^{\mathrm{x}}$.

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One solution is obvious - x=1. Prove that there are no other solutions using the monotonic increase of some functions.

## Solution

One of the solutions can be guessed immediately: $\mathrm{x}=1$. Now it is enough to show that this equation has no more than one solution. Transform the equation to the form $(2 / 5)^{\mathrm{x}}+(3 / 5)^{\mathrm{x}}=1$. On the left side, we have a function that is decreasing over the entire number line (as the sum of two decreasing functions). Therefore, this function takes each real value no more than once. Hence, for any number c, the equation $(2 / 5)^{\mathrm{x}}+(3 / 5)^{\mathrm{x}}=c$ has no more than one solution.

## Answer

$\mathrm{x}=1$.