Problem 67 - Introductory and Intermediate Algebra for College Students 4th

Question

Use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. $$\log _{\pi} 63$$

Step-by-Step Solution

Answer

The value of \(\log _{\pi} 63\) rounded to four decimal places is 1.3423 after using the change-of-base formula and a calculator.

Step 1: Apply the change-of-base formula

Transform \(\log _{\pi} 63\) using the change-of-base formula. We can convert this into the form \(\frac{\log_c b}{\log_c a}\). Choose either common logarithms or natural logarithms as the new base (indicated by \(c\)). For this example, let's choose the common logarithm (base 10). So, it becomes \(\frac{\log_{10} 63}{\log_{10} \pi}\).

Step 2: Calculate the logarithms

Now, we need to calculate \(\log_{10} 63\) and \(\log_{10} \pi\). Use a calculator to find these values. It's important to remember to round these to four decimal places.

Step 3: Divide the results

Finally, divide the result of \(\log_{10} 63\) by the result of \(\log_{10} \pi\) to obtain the value of \(\log _{\pi} 63\). This value, the result of the division, should also be rounded to four decimal places.