Example 16. Solve the equation

$$
\log _{4}(x+3)+\log _{4}(x-1)=2-\log _{4} 8
$$
Solution. The domain of the equation is determined by the system

$$
\left\{\begin{array}{l}
x+3>0 \\
x-1>0
\end{array}\right.
$$

the solution of which is the interval $11$, i.e., wider than the domain of the equation (12).

The solution of equation (12) can be briefly written as:

$$
\begin{aligned}
& \Leftrightarrow\left\{\begin{array} { l } 
{ x + 3 > 0 , } \\
{ x - 1 > 0 , } \\
{ \operatorname { l o g } _ { 4 } ( x + 3 ) ( x - 1 ) = \operatorname { l o g } _ { 4 } 2 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
x>-3 \\
x>1, \\
(x+3)(x-1)=2
\end{array} \Leftrightarrow\right.\right. \\
& \Leftrightarrow\left\{\begin{array} { l } 
{ x > 1 , } \\
{ x ^ { 2 } + 2 x - 5 = 0 }
\end{array} \Leftrightarrow \left\{\begin{array}{l}
x>1, \\
(x+1+\sqrt{6})(x-(\sqrt{6}-1))=0 \\
\Leftrightarrow x=\sqrt{6}-1 .
\end{array}\right.\right.
\end{aligned}
$$

Thus, the number $\sqrt{\overline{6}}-1$ is the only root of equation (12).

Transforming the equation using formulas $1^{0}-4^{0}$ from "left to right" often leads to an equation with a narrower domain than the original equation, and, consequently, possible loss of roots of the original equation. To avoid losing roots, more general formulas should be used.

More general logarithmic formulas. Let $f$ and $g$ be some functions and let $a>0, a \neq 1$. Then, if $f \neq 0$ and $g \neq 0$; then

1. $\log _{a}(f g)=\log _{a}|f|+\log _{a}|g|$.
2. $\log _{a} \frac{f}{g}=\log _{a}|f|-\log _{a}|g|$.

$$
\text { 3. } \log _{a} f^{2 \alpha}=2 \alpha \log _{a}|f| . \quad 4 . \log _{a^{2 \beta}} f^{2 \alpha}=\frac{\alpha}{\beta} \log _{|a|}|f|
$$

The left and right sides of formula 1 have different restrictions on $f$ and $g$: the left side is meaningful when $f$ and $g$ have the same sign, while the right side is meaningful for any $f$ and $g$ different from zero. Similarly, the domain of the right side of formulas $2-4$ may be wider than the domain of the left side.

Therefore, transforming the equation using formulas $1-4$ from "left to right" (as they are written) will lead to an equation that is a consequence of the original equation. Transforming the equation with formal use of formulas $1-4$ from "right to left" may result in the loss of roots.