In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places.logπ 63

Verified step by step guidance

1. insert step 1> Convert the logarithm with base \( \pi \) to a natural logarithm using the change of base formula: \( \log_{\pi} 63 = \frac{\ln 63}{\ln \pi} \).
2. insert step 2> Use a calculator to find \( \ln 63 \).
3. insert step 3> Use a calculator to find \( \ln \pi \).
4. insert step 4> Divide the result from step 2 by the result from step 3 to evaluate \( \log_{\pi} 63 \).
5. insert step 5> Round the result to four decimal places.

Key Concepts

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in equations of the form b^x = y. The logarithm log_b(y) answers the question: 'To what power must the base b be raised to obtain y?' Understanding logarithms is essential for evaluating expressions like log_π(63), where π is the base.

Change of Base Formula

The Change of Base Formula allows us to convert logarithms from one base to another, which is particularly useful when the base is not easily calculable. The formula states that log_b(a) = log_k(a) / log_k(b) for any positive k. This is crucial for evaluating log_π(63) using common (base 10) or natural (base e) logarithms, as calculators typically do not have a π button.

Common and Natural Logarithms

Common logarithms (log) use base 10, while natural logarithms (ln) use base e (approximately 2.718). Both types of logarithms are widely used in mathematics and science. When evaluating log_π(63), one can use either common or natural logarithms in conjunction with the Change of Base Formula to find the value to four decimal places.