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

Introduction to Logarithms

College Algebra

6. Exponential & Logarithmic Functions

Introduction to Logarithms

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1:38 minutes

Problem 75

Textbook Question

In Exercises 75–80, find the domain of each logarithmic function. f(x) = log5 (x+4)

Verified step by step guidance

Identify the expression inside the logarithm: \( x + 4 \).

Recall that the argument of a logarithm must be greater than zero: \( x + 4 > 0 \).

Solve the inequality for \( x \): \( x > -4 \).

The domain of the function is all \( x \) values that satisfy the inequality.

Express the domain in interval notation: \((-4, \infty)\).

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Key Concepts

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

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined for positive real numbers. The general form is f(x) = log_b(x), where b is the base and x must be greater than zero. Understanding the properties of logarithms is essential for determining their domains.

Domain of a Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. For logarithmic functions, the argument of the logarithm must be positive. This means that to find the domain, we need to set the argument greater than zero and solve for x.

Inequalities

Inequalities are mathematical expressions that show the relationship between two values, indicating that one is greater than, less than, or equal to the other. In the context of finding the domain of logarithmic functions, we often set up an inequality based on the function's argument and solve it to find the valid range of x.