1. (2 points) Solve the system:

$$
\left\{\begin{array}{l}
\left|\frac{\ln x}{\ln 2}\right|+\frac{4 x^{2}}{15}-\frac{16}{15}=0 \\
\frac{\ln \left(x+\frac{2}{3}\right)}{\ln 7}+12 x-5=0
\end{array}\right.
$$
Answer: No solutions.

## Solution:

Consider the second equation of the system. For $x \geqslant \frac{5}{12}$, the logarithm is positive, and $12 x-5$ is non-negative. For $x \leqslant \frac{1}{3}$, the logarithm is non-positive, and $12 x-5$ is negative. Therefore, all roots of this equation, however many there are, lie between the numbers $\frac{1}{3}$ and $\frac{5}{12}$.

On this interval, $\ln x < 0$, so the first equation takes the form

$$
\frac{\ln x}{\ln 2} = \frac{4 x^{2}}{15} - \frac{16}{15}
$$

This equation has no more than two roots, since the function on the right side is convex, while the function on the left side is concave (graphical reasoning is allowed). Moreover, one of the roots of this equation is greater than 2, since at $x=2$ the left side is greater than the right, and, for example, at $x=1024$ the opposite is true $\left(10 < \frac{4}{15} \cdot (1024^{2} - 16)\right)$. The smaller root of this equation is $\frac{1}{2}$. Thus, none of these roots fall within the specified interval.

Alternatively, one can notice that between the numbers $\frac{1}{3}$ and $\frac{5}{12}$, the function $\frac{\ln x}{\ln 2} - \frac{4 x^{2}}{15} + \frac{16}{15}$ is decreasing (the derivative is negative), and the values of the function at the boundaries of the interval are of the same sign (less than $0$), so the function has no roots.