Task B-3.1. Solve the equation

$$
\log _{5 x-2} 2+2 \cdot \log _{5 x-2} x=\log _{5 x-2}(x+1)
$$
Solution. Let's determine the conditions for which the given equation makes sense.

$$
\begin{aligned}
& 5 x-2>0, \quad 5 x-2 \neq 1, \quad x>0, \quad x+1>0 . \\
& \log _{5 x-2} 2+2 \cdot \log _{5 x-2} x=\log _{5 x-2}(x+1) \\
& \log _{5 x-2}\left(2 x^{2}\right)=\log _{5 x-2}(x+1) \\
& 2 x^{2}=x+1
\end{aligned}
$$

The solutions to this quadratic equation are $x_{1}=1, x_{2}=-\frac{1}{2}$.

However, only $x_{1}=1$ satisfies the condition, so it is the only solution.