1. Given a positive real number $a$ satisfies $a^{a}=(9 a)^{8 a}$, then the value of $\log _{a}(3 a)$ is
$$
\begin{array}{l}
(9 a)^{8 a}=a^{a} \Rightarrow\left[(9 a)^{8}\right]^{a}=a^{a} \Rightarrow(9 a)^{8}=a \Rightarrow 8 \log _{a}(9 a)=1 \\
\Rightarrow 8 \log _{a} 9+8=1 \Rightarrow 16 \log _{a} 3+16=9 \Rightarrow \log _{a}(3 a)=\frac{9}{16} .
\end{array}
$$

The translation is as follows:

$$
\begin{array}{l}
(9 a)^{8 a}=a^{a} \Rightarrow\left[(9 a)^{8}\right]^{a}=a^{a} \Rightarrow(9 a)^{8}=a \Rightarrow 8 \log _{a}(9 a)=1 \\
\Rightarrow 8 \log _{a} 9+8=1 \Rightarrow 16 \log _{a} 3+16=9 \Rightarrow \log _{a}(3 a)=\frac{9}{16} .
\end{array}
$$