### 7.301 The equation $4^{x}+10^{x}=25^{x}$ has a unique root.

Find it and compare it with 0 and 1.
## Solution.

$$
\begin{aligned}
& 2^{2 x}+2^{x} \cdot 5^{x}=5^{2 x} \Leftrightarrow\left(\frac{2}{5}\right)^{2 x}+\left(\frac{2}{5}\right)^{x}-1=0, \Leftrightarrow\left(\frac{2}{5}\right)^{x}=\frac{-1+\sqrt{5}}{2} \Leftrightarrow \\
& \Leftrightarrow x=\log _{0.4} \frac{\sqrt{5}-1}{2} \text {. Since } \frac{2}{5}\log _{0.4} \frac{\sqrt{5}-1}{2}>\log _{0.4} 1 \Rightarrow 1>x>0
\end{aligned}
$$

Answer: $x=\log _{0.4} \frac{\sqrt{5}-1}{2}, 0<x<1$.