4. Solve the equation (for $x$)

$$
\log _{a x} a+3 \log _{a^{2} x} a=0, \text { if } a>1
$$
Solution. We have

$$
\begin{aligned}
\log _{a x} a + 3 \log _{a}^{2} x & =\frac{1}{\log _{a} a x}+\frac{3}{\log _{a} a^{2} x} \\
& =\frac{\log _{a} a^{2} x + 3 \log _{a} a x}{\log _{a} a x \cdot \log _{a} a^{2} x} \\
& =\frac{\log _{a} a^{5} x^{3}}{\log _{a} a x \cdot \log _{a} a^{2} x}
\end{aligned}
$$

where $x \neq \frac{1}{a}$ and $x \neq \frac{1}{a^{2}}$. Therefore, the given equation is equivalent to the equation

$$
\frac{\log _{a} a^{5} x^{3}}{\log _{a} a x \cdot \log _{a} a^{2} x}=0
$$

from which we obtain $\log _{a} a^{5} x^{3}=0$, i.e., $x=\frac{1}{a \sqrt[4]{a}}$.

## 4th Year