smc

The solutions imply that finding the ratio $\frac{a}{b}$ will solve the problem. We compute $\frac{a}{b}$, use change-of-base to a neutral base, rearrange the terms, and then use the reverse-change-of-base: $\frac{{\log }_{8}225}{{\log }_{2}15}$ $\frac{\frac{\ln 225}{\ln 8}}{\frac{\ln 15}{\ln 2}}$ $\frac{\ln 225\ln 2}{\ln 15\ln 8}$ $\frac{\ln 225}{\ln 15}\cdot \frac{\ln 2}{\ln 8}$ ${\ln }_{15}225\cdot {\ln }_{8}2$ Since $15^2=225$, the first logarithm is $2$. Since $8^{\frac{1}{3}}=2$, the second logarithm is $\frac{1}{3}$. Thus, we have $\frac{a}{b}=\frac{2}{3}$, or $a=\frac{2}{3}b$, which is option $\boxed{a=2b/3}$.