Use the change-of-base formula to approximate the logarithm accurate to the nearest ten-thousandth.
log3  3/4 

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.

Use the change-of-base formula to approximate the logarithm accurate to the nearest ten-thousandth.
log3  3/4

Learn how to approximate the logarithm log3(3/4) to the nearest ten-thousandth using the change-of-base formula. This step-by-step solution covers key logarithmic concepts, including base-10 conversions and accuracy up to four decimal places. Suitable for high school students in Grades 10-12.

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