5. Let $a, b, c>1$, satisfying $\left(a^{2} b\right)^{\log _{a} c}=a \cdot(a c)^{\log _{a} b}$, then the value of $\log _{c}(a b)$ is

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$$
\begin{array}{l}
\log _{a} c\left(2+\log _{a} b\right)=1+\log _{a} b\left(1+\log _{a} c\right) \Rightarrow 2 \log _{a} c=1+\log _{a} b=\log _{a}(a b) \\
\Rightarrow c^{2}=a b \Rightarrow \log _{c}(a b)=2 .
\end{array}
$$