7.303 Expression

$T=\log _{m} A \cdot \log _{n} A+\log _{n} A \cdot \log _{p} A+\log _{p} A \cdot \log _{m} A$ represent in the form of a product.
Solution.

$$
\begin{aligned}
& T=\frac{1}{\log _{A} m \cdot \log _{A} n}+\frac{1}{\log _{A} n \cdot \log _{A} p}+\frac{1}{\log _{A} p \cdot \log _{A} m}=\frac{\log _{A} p+\log _{A} m+\log _{A} n}{\log _{A} m \cdot \log _{A} n \cdot \log _{A} p}= \\
& =\log _{A}(m \cdot n \cdot p) \cdot \log _{p} A \cdot \log _{n} A \cdot \log _{m} A
\end{aligned}
$$