We use the change-of-base formula to compute $\log_3 \frac{3}{4}$:

$\log_3 \frac{3}{4} = \frac{\log_{10} \frac{3}{4}}{\log_{10} 3}$

Let's calculate each component using base-10 logarithms.

1. Calculate $\log_{10} \frac{3}{4}$: $\log_{10} \frac{3}{4} = \log_{10} 0.75 \approx -0.1250$

2. Calculate $\log_{10} 3$: $\log_{10} 3 \approx 0.4771$

3. Now, divide the two results: $\log_3 \frac{3}{4} = \frac{-0.1250}{0.4771} \approx -0.2619$

Thus, $\log_3 \frac{3}{4} \approx -0.2619$.

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Would you like any further details or have any questions? Here are some related questions to consider:

1. How does the change-of-base formula work with natural logarithms?
2. Can you apply this method to approximate $\log_3 5$?
3. What is the purpose of using the change-of-base formula in logarithmic calculations?
4. How would rounding to different decimal places affect the accuracy of your result?
5. Can you use the change-of-base formula for non-numerical or symbolic expressions?

Tip: Use a scientific calculator or software to ensure accuracy when dealing with logarithms to many decimal places.