7.55 Find $\log _{30} 8$, given that $\lg 5=a$ and $\lg 3=b$.

Solve the equations (7.56-7.74):
7.55 Using formulas (7.6), (7.7), (7.4), $(7.5)$ sequentially, we get

$$
\log _{30} 8=3 \log _{30} 2=\frac{3 \lg 2}{\lg 30}=\frac{3 \lg \frac{10}{5}}{\lg (3 \cdot 10)}=\frac{3(1-\lg 5)}{\lg 3+1}=\frac{3(1-a)}{b+1}
$$

Answer: $\frac{3(1-a)}{b+1}$.