The problem in the image asks to use the change-of-base theorem to evaluate $\log_{\pi}{37}$, and round the result to four decimal places.

Change-of-base formula:

$\log_b{a} = \frac{\log_c{a}}{\log_c{b}}$ where $c$ can be any base (usually base 10 or natural logarithm, base $e$).

Here, we are solving for $\log_{\pi}{37}$, so using base 10 logarithms, we apply the formula: $\log_{\pi}{37} = \frac{\log{37}}{\log{\pi}}$

Now, let's calculate this step-by-step:

1. Find $\log{37}$ (base 10).
2. Find $\log{\pi}$ (base 10).
3. Divide the two results.
4. Round the final answer to four decimal places.

Step-by-step calculations:

• $\log{37} \approx 1.5682$
• $\log{\pi} \approx 0.4971$

Now, divide: $\frac{1.5682}{0.4971} \approx 3.1553$

Thus, $\log_{\pi}{37} \approx 3.1553$ when rounded to four decimal places.

Would you like more details or have any other questions?

Here are 5 related questions you might find useful:

1. How can the change-of-base theorem be applied with natural logarithms?
2. What is the difference between common and natural logarithms?
3. How can you simplify expressions involving logarithms with different bases?
4. Why does the change-of-base formula work for any new base?
5. How do logarithmic properties help in solving exponential equations?

Tip: Always keep track of decimal places during intermediate steps to avoid rounding errors early on!