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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2:04 minutes

Problem 31

Textbook Question

Solve each equation. log￬4 x = 3

Verified step by step guidance

Understand that the equation \( \log_4 x = 3 \) is asking for the value of \( x \) such that when 4 is raised to the power of 3, it equals \( x \).

Rewrite the logarithmic equation in its exponential form: \( x = 4^3 \).

Calculate \( 4^3 \) by multiplying 4 by itself three times: \( 4 \times 4 \times 4 \).

Simplify the expression to find the value of \( x \).

Verify your solution by substituting \( x \) back into the original equation to ensure that \( \log_4 x = 3 \) holds true.

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Key Concepts

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

Logarithms

A logarithm is the inverse operation to exponentiation, representing the power to which a base must be raised to obtain a given number. In the equation log₄ x = 3, it indicates that 4 raised to the power of 3 equals x. Understanding logarithms is essential for solving equations involving them.

Exponential Functions

Exponential functions are mathematical functions of the form f(x) = a^x, where 'a' is a positive constant. They are crucial for understanding how logarithms work, as they describe the relationship between the base and the exponent. In this case, knowing that 4^3 = 64 helps in solving the logarithmic equation.

Change of Base Formula

The change of base formula allows the conversion of logarithms from one base to another, facilitating easier calculations. It states that logₐ b = logₓ b / logₓ a for any positive x. This concept can be useful if you need to evaluate logarithms with bases that are not easily computable.