7.1. How many values of the parameter $a$ exist for which the equation

$$
4 a^{2}+3 x \lg x+3 \lg ^{2} x=13 a \lg x+a x
$$

has a unique solution
Answer: 2.

Solution. The given equation is equivalent to the equation $(3 \lg x-a)(x+\lg x-4 a)=0$, from which we have the system

$$
\left[\begin{array}{l}
\lg x=\frac{a}{3} \\
x+\lg x=4 a
\end{array}\right.
$$

Each equation in this system has a unique positive solution for any $a$, since the continuous functions $f_{1}(x)=\lg x$ and $f_{2}(x)=x+\lg x$ are strictly increasing on their domain $(0 ;+\infty)$ and take all possible values from $(-\infty ;+\infty)$. Therefore, it is necessary for these solutions to coincide. Then $x=11 a / 3, \lg x=a / 3$, hence, $10^{a / 3}=11 a / 3$. This equation has exactly two solutions due to the fact that the function $g(t)=10^{t}-11 t$ is strictly decreasing on $(-\infty ; \lg (11 \lg e)]$, strictly increasing on $[\lg (11 \lg e) ;+\infty)$, takes a negative value at the point $t=1$, and positive values at the points $t=0$ and $t=2$.