1. Given positive real numbers $a, b$ satisfy $a^{b}=(8 a)^{9 b}$, then the value of $\log _{a}\left(4 a^{2}\right)$ is $\qquad$
$$
\begin{array}{l}
a^{b}=\left[(8 a)^{9}\right]^{b} \Rightarrow a=(8 a)^{9}=8^{9} a^{9} \Rightarrow a^{8}=8^{-9}=2^{-27} \Rightarrow 8=-27 \log _{a} 2 \\
\Rightarrow \log _{a} 2=-\frac{8}{27} \Rightarrow \log _{a} 4=-\frac{16}{27}, \text { so } \log _{a}\left(4 a^{2}\right)=2+\log _{a} 4=\frac{38}{27} .
\end{array}
$$