Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. log_π e

Verified step by step guidance

1. Recall the change-of-base formula for logarithms: \(\log_a b = \frac{\log_c b}{\log_c a}\), where \(c\) is any positive number different from 1. Common choices for \(c\) are 10 (common logarithm) or \(e\) (natural logarithm).
2. Apply the change-of-base formula to \(\log_{\pi} e\) by choosing the natural logarithm base \(e\): \(\log_{\pi} e = \frac{\ln e}{\ln \pi}\).
3. Evaluate the numerator \(\ln e\). Since the natural logarithm of \(e\) is 1, this simplifies the expression to \(\frac{1}{\ln \pi}\).
4. Calculate \(\ln \pi\) using a calculator or logarithm table to get a decimal approximation.
5. Divide 1 by the decimal approximation of \(\ln \pi\) to find the approximate value of \(\log_{\pi} e\), then round your answer to four decimal places.

Key Concepts

Logarithms and Their Bases

A logarithm answers the question: to what power must the base be raised to produce a given number? The notation log_b a means the logarithm of a with base b. Understanding how different bases affect the value of logarithms is essential for manipulating and approximating them.

Change-of-Base Theorem

The change-of-base theorem allows you to rewrite a logarithm with any base into a ratio of logarithms with a more convenient base, typically base 10 or e. It states that log_b a = (log_c a) / (log_c b), where c is a new base. This is useful for calculating logarithms on calculators that only support certain bases.

Approximation of Logarithms Using Calculators

Since calculators usually provide logarithms in base 10 or base e (natural logs), approximating logarithms with other bases requires using the change-of-base formula. After rewriting, you compute the numerator and denominator logarithms and divide, then round the result to the desired decimal places.