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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1:19 minutes

Problem 1

Textbook Question

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.log5 (7 × 3)

Verified step by step guidance

Identify the logarithmic expression: $ \log_5 (7 \times 3) $.

Apply the product rule of logarithms: $ \log_b (MN) = \log_b M + \log_b N $.

Expand the expression using the product rule: $ \log_5 7 + \log_5 3 $.

Each term is now a separate logarithm with base 5.

The expression is fully expanded as $ \log_5 7 + \log_5 3 $.

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Key Concepts

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

Properties of Logarithms

The properties of logarithms are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule, which states that the logarithm of a product is the sum of the logarithms of the factors (log_b(mn) = log_b(m) + log_b(n)), the quotient rule for division, and the power rule for exponents. Understanding these properties is essential for expanding and simplifying logarithmic expressions.

Logarithmic Expansion

Logarithmic expansion involves breaking down a logarithmic expression into simpler components using the properties of logarithms. For example, the expression log_b(mn) can be expanded to log_b(m) + log_b(n). This process is crucial for solving logarithmic equations and simplifying complex expressions, making it easier to evaluate or manipulate them.

Base of a Logarithm

The base of a logarithm indicates the number that is raised to a power to obtain a given value. In the expression log_b(a), 'b' is the base, and 'a' is the argument. Understanding the base is important because it determines the scale and behavior of the logarithmic function, influencing how we interpret and evaluate logarithmic expressions.