7.157. It is known that $\log _{a} x=\alpha, \log _{b} x=p, \log _{c} x=\gamma, \log _{d} x=\delta$ and $x \neq 1$. Find $\log _{a b c d} x$.
Solution.

$$
\begin{aligned}
& \log _{a b c d} x=\frac{\log _{x} x}{\log _{x} a b c d}=\frac{1}{\log _{x} a+\log _{x} b+\log _{x} c+\log _{x} d}= \\
& =\frac{1}{\frac{1}{\log _{a} x}+\frac{1}{\log _{b} x}+\frac{1}{\log _{c} x}+\frac{1}{\log _{d} x}}= \\
& =\frac{1}{\frac{1}{\alpha}+\frac{1}{\beta}+\frac{1}{\gamma}+\frac{1}{\delta}}=\frac{\alpha \beta \gamma \delta}{\beta \gamma \delta+\alpha \gamma \delta+\alpha \beta \delta+\alpha \beta \gamma} .
\end{aligned}
$$

Answer: $\frac{\alpha \beta \gamma \delta}{\beta \gamma \delta+\alpha \gamma \delta+\alpha \beta \delta+\alpha \beta \gamma}$.