We are asked to simplify the expression $\frac{\log 9 - \log 8}{\log 81 - \log 64}$.

1. Problem Description

We are asked to simplify the expression $\frac{\log 9 - \log 8}{\log 81 - \log 64}$.

2. Solution Steps

First, we use the logarithm property $\log a - \log b = \log \frac{a}{b}$.

Therefore,

$\log 9 - \log 8 = \log \frac{9}{8}$

$\log 81 - \log 64 = \log \frac{81}{64}$

Then, the expression becomes

$\frac{\log \frac{9}{8}}{\log \frac{81}{64}}$

Now, we can rewrite the fraction $\frac{81}{64}$ as $\frac{9^2}{8^2} = (\frac{9}{8})^2$.

So, the expression becomes

$\frac{\log \frac{9}{8}}{\log (\frac{9}{8})^2}$

Using the logarithm property $\log a^b = b \log a$, we have $\log (\frac{9}{8})^2 = 2 \log \frac{9}{8}$.

Therefore,

$\frac{\log \frac{9}{8}}{\log (\frac{9}{8})^2} = \frac{\log \frac{9}{8}}{2 \log \frac{9}{8}} = \frac{1}{2}$.

3. Final Answer

$\frac{1}{2}$