Example 15. Solve the equation

$$
\log _{1 / 3}\left(x^{2}+3 x-4\right)=\log _{1 / 3}(2 x+2)
$$
Solution. Solving the system of inequalities

$$
\left\{\begin{array}{l}
x^{2}+3 x-4>0 \\
2 x+2>0
\end{array}\right.
$$

we find the domain of definition (ODZ) of equation (10): $x>1$.

Potentiating equation (10), we obtain the equation

$$
x^{2}+3 x-4=2 x+2
$$

which is a consequence of it.

Equation (11) has two roots: $x_{1}=2$ and $x_{2}=-3$. The number $x_{1}=2$ falls within the ODZ of equation (10) and is therefore its root. The number $x_{2}=-3$ does not fall within the ODZ of equation (10) and is thus an extraneous root.

Thus, the number 2 is the only root of equation (10).

Formulas of logarithms. Let $f$ and $g$ be some functions and $a>0, a \neq 1$. Then, if $f>0$ and $g>0$, we have

$1^{\circ} . \log _{a}(f g)=\log _{a} f+\log _{a} g$.

2.. $\log _{a} \frac{f}{g}=\log _{a} f-\log _{a} g$.

$3^{\circ} . \log _{a} f^{\alpha}=\alpha \log _{a} f$.

$4^{\circ} . \log _{a^{\beta}} f^{\alpha}=\frac{\alpha}{\beta} \log _{a} f$.

Applying any of these formulas formally (without considering the inequalities $f>0$ and $g>0$), one should keep in mind that the domains of definition (ODZ) of the left and right parts of each of them may be different. For example, the expression $\log _{a} f+\log _{a} g$ is defined for $f>0$ and $g>0$, while the expression $\log _{a}(f g)$ is defined both for $f>0, g>0$ and for $f<0, g<0$. Similarly, the ODZ of the left part of formulas $2^{\circ}-4^{\circ}$ may be wider than the ODZ of the right part.

Thus, transforming an equation by formally using formulas $1^{\circ}-4^{\circ}$ "from right to left" leads to an equation that is a consequence of the original equation (its ODZ may be wider). In this case, extraneous roots of the original equation may appear; therefore, at the end of the solution, one must check the membership of each root of the last equation to the ODZ of the original equation.