Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

6. Exponential & Logarithmic Functions

Properties of Logarithms

College Algebra

6. Exponential & Logarithmic Functions

Properties of Logarithms

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2:45 minutes

Problem 85

Textbook Question

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

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 \( e \).

Apply the change-of-base formula to \( \log_{\pi} e \) using base 10: \( \log_{\pi} e = \frac{\log_{10} e}{\log_{10} \pi} \).

Alternatively, apply the change-of-base formula using base \( e \) (natural logarithm): \( \log_{\pi} e = \frac{\ln e}{\ln \pi} \).

Recall that \( \ln e = 1 \), simplifying the expression to \( \frac{1}{\ln \pi} \) if using natural logarithms.

Use a calculator to find \( \log_{10} e \) and \( \log_{10} \pi \) or \( \ln \pi \) to approximate the value to four decimal places.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.