7.154. $\frac{\log _{a} b-\log _{\sqrt{a} / b^{3}} \sqrt{b}}{\log _{a / b^{4}} b-\log _{a / b^{6}} b}: \log _{b}\left(a^{3} b^{-12}\right)$.
## Solution.

$$
\begin{aligned}
& \frac{\log _{a} b-\log _{\sqrt{a} / b^{3}} \sqrt{b}}{\log _{a / b^{4}} b-\log _{a / b^{6} b}}: \log _{b}\left(a^{3} b^{-12}\right)=\frac{\log _{a} b-\frac{\log _{a} \sqrt{b}}{\log _{a} \frac{\sqrt{a}}{b^{3}}}}{\frac{\log _{a} b}{\log _{a} \frac{a}{b^{4}}}-\frac{\log _{a} b}{\log _{a} \frac{a}{b^{6}}}}: \frac{\log _{a} a^{3} \cdot b^{-12}}{\log _{a} b}= \\
& =\frac{\log _{a} b-\frac{\frac{1}{2} \log _{a} b}{\frac{1}{2}-3 \log _{a} b}}{\frac{\log _{a} b}{1-4 \log _{a} b}-\frac{\log _{a} b}{1-6 \log _{a} b}} \cdot \frac{\log _{a} b}{3-12 \log _{a} b}= \\
& =\frac{-3 \log _{a}^{2} b\left(1-4 \log _{a} b\right)\left(1-6 \log _{a} b\right)}{\left(-6 \log _{a}^{2} b+4 \log _{a}^{2} b\right)\left(\frac{1}{2}-3 \log _{a} b\right)} \cdot \frac{\log _{a} b}{3\left(1-4 \log _{a} b\right)}=\log _{a} b
\end{aligned}
$$

Answer: $\log _{a} b$.