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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2:11 minutes

Problem 79

Textbook Question

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. log_2 5

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 \( e \).

Choose a new base for the logarithm. Common choices are base 10 (common logarithm) or base \( e \) (natural logarithm). For this example, we'll use base 10.

Apply the change-of-base formula: \( \log_2 5 = \frac{\log_{10} 5}{\log_{10} 2} \).

Use a calculator to find \( \log_{10} 5 \) and \( \log_{10} 2 \).

Divide the result of \( \log_{10} 5 \) by \( \log_{10} 2 \) to find the approximation of \( \log_2 5 \) 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 compute logarithms with bases other than 10 or e by converting them into a more manageable form. Specifically, it states that log_b(a) can be expressed as log_k(a) / log_k(b) for any positive k. This is particularly useful when using calculators that typically only compute common (base 10) or natural (base e) logarithms.

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 the Change of Base Theorem, approximation techniques may be necessary to achieve a desired level of precision. This often involves using a calculator to find the values of the logarithms in the numerator and denominator, and then performing the division to obtain the final result. Rounding to four decimal places ensures that the answer is both precise and practical for applications.