Problem 2. Solve in the set of real numbers the equation $x+\log _{2}\left(1+\sqrt{\frac{5^{x}}{3^{x}+4^{x}}}\right)=4+\log _{1 / 2}\left(1+\sqrt{\frac{25^{x}}{7^{x}+24^{x}}}\right)$.
Solution. Note that $x=2$ is a solution.

1 point

We will show that this solution is unique. For $x>2$ we have $(3 / 5)^{x}+(4 / 5)^{x}<1$, so the left-hand side is strictly greater than 3.

On the other hand, $(7 / 25)^{x}+(24 / 25)^{x}<1$ and $\log _{1 / 2}\left(1+\sqrt{25^{x} /\left(7^{x}+24^{x}\right)}\right)<\log _{1 / 2} 2=-1$. It follows that the right-hand side is strictly less than 3, so the equation has no solutions in the set $(2, \infty)$.