To evaluate the expression given in the image:

$\frac{81 \log_{9} 3}{36 \log_{9} 6}$

we can simplify this step-by-step without using a calculator, focusing on properties of logarithms and the values involved.

Step 1: Simplify $81 \log_9 3$

Since $\log_9 3$ is the logarithm with base 9 of 3, we can rewrite it in terms of base conversion if needed, but first, let's evaluate the constants.

• Notice that $9 = 3^2$, which implies that $\log_9 3 = \frac{1}{2}$, because $9^{\frac{1}{2}} = 3$.
• Therefore, $81 \log_9 3 = 81 \times \frac{1}{2} = 40.5$.

Step 2: Simplify $36 \log_9 6$

Similarly, $\log_9 6$ would require us to evaluate how 6 relates to the base 9. We can write $6$ in terms of powers of 3 for simplicity.