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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1:07 minutes

Problem 41

Textbook Question

In Exercises 41–70, use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator.log 5 + log 2

Verified step by step guidance

Identify the property of logarithms that can be used to combine the terms. The property is: \( \log_b M + \log_b N = \log_b (M \cdot N) \).

Apply the property to the given expression: \( \log 5 + \log 2 \).

Combine the logarithms into a single logarithm: \( \log (5 \cdot 2) \).

Simplify the expression inside the logarithm: \( \log 10 \).

The expression is now condensed into a single logarithm with a coefficient of 1: \( \log 10 \).

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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 sum of two logarithms with the same base can be expressed as the logarithm of the product of their arguments (log_a(b) + log_a(c) = log_a(bc). This property is essential for condensing logarithmic expressions.

Condensing Logarithmic Expressions

Condensing logarithmic expressions involves combining multiple logarithms into a single logarithm. This is achieved by applying the properties of logarithms, particularly the product, quotient, and power rules. For example, in the expression log 5 + log 2, you can condense it to log(5 * 2) = log 10, demonstrating the application of the product rule.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions means finding the numerical value of a logarithm. For instance, log 10 equals 1 because 10 raised to the power of 1 equals 10. Understanding how to evaluate logarithmic expressions is crucial, especially when simplifying or condensing them, as it allows for a clearer understanding of the resulting expressions.