Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. log_π e

Verified step by step guidance

1. Identify the change-of-base formula: \( \log_b a = \frac{\log_c a}{\log_c b} \), where \( c \) is a new base, typically 10 or \( e \).
2. Apply the change-of-base formula to \( \log_{\pi} e \) using base 10: \( \log_{\pi} e = \frac{\log_{10} e}{\log_{10} \pi} \).
3. Alternatively, apply the change-of-base formula using base \( e \) (natural logarithm): \( \log_{\pi} e = \frac{\ln e}{\ln \pi} \).
4. Recall that \( \ln e = 1 \), simplifying the expression to \( \frac{1}{\ln \pi} \) if using natural logarithms.
5. Use a calculator to find \( \log_{10} e \) and \( \log_{10} \pi \) or \( \ln \pi \) to approximate the value to four decimal places.

Key Concepts

Change of Base Theorem

The Change of Base Theorem allows us to convert logarithms from one base to another. It states that for any positive numbers a, b, and c (where a and b are not equal to 1), the logarithm can be expressed as log_b(a) = log_c(a) / log_c(b). This theorem is particularly useful when calculating logarithms with bases that are not easily computable using standard calculators.

Logarithm Basics

A logarithm is the inverse operation to exponentiation, answering the question: to what exponent must a base be raised to produce a given number? For example, log_b(a) = c means that b^c = a. Understanding the properties of logarithms, such as the product, quotient, and power rules, is essential for manipulating and solving logarithmic expressions.

Approximation Techniques

When calculating logarithms, especially with non-standard bases, approximation techniques may be necessary. This often involves using a scientific calculator or logarithm tables to find values to a specified number of decimal places. In this context, approximating log_π e requires converting it to a more manageable base, such as 10 or e, to facilitate the calculation.