Use the​ change-of-base formula and a calculator to evaluate the following logarithm.

log Subscript pi Baseline 43logπ43

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Part 1

log Subscript pi Baseline 43logπ43almost equals≈enter your response here

​(Round the final answer to four decimal places as needed. Round all intermediate values to five decimal places as​ needed.)

To evaluate \( \log_{\pi} 43 \) using the change-of-base formula, we proceed as follows:

### Step-by-Step Solution

1. **Apply the Change-of-Base Formula:**
   The change-of-base formula for logarithms is:
   \[
   \log_a b = \frac{\log_c b}{\log_c a}
   \]
   Here, we want to evaluate \( \log_{\pi} 43 \), so we can use base 10 or base \( e \) (natural logarithm). Let's use base 10 for simplicity.

   Thus:
   \[
   \log_{\pi} 43 = \frac{\log_{10} 43}{\log_{10} \pi}
   \]

2. **Calculate \( \log_{10} 43 \) and \( \log_{10} \pi \):**
   Using a calculator, we get:
   - \( \log_{10} 43 \approx 1.63347 \) (rounded to five decimal places)
   - \( \log_{10} \pi \approx 0.49715 \) (rounded to five decimal places)

3. **Divide the Results:**
   \[
   \log_{\pi} 43 \approx \frac{1.63347}{0.49715} \approx 3.2857
   \]

### Final Answer:
\[
\log_{\pi} 43 \approx 3.2857
\]

Would you like any additional steps explained in more detail, or do you have any further questions?

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1. What is the change-of-base formula, and how is it derived?
2. Why does rounding intermediate values affect the final result?
3. How would the result change if we used a different logarithmic base in the calculation?
4. Can the change-of-base formula be used for any base of a logarithm?
5. How is \( \pi \) typically approximated in logarithmic calculations?

**Tip:** Always round intermediate values to one more decimal place than needed in the final answer for accuracy.

Use the change-of-base formula and a calculator to evaluate the following logarithm. log_π 43. Round the final answer to four decimal places as needed. Round all intermediate values to five decimal places as needed.

Learn how to evaluate log base π of 43 using the change-of-base formula. This problem involves applying the change-of-base formula and using a calculator to round values accurately. Suitable for high school students in Grades 10-12.

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