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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2:27 minutes

Problem 41

Textbook Question

Solve each equation. 3x - 15 = log￬x 1 (x>0, x≠1)

Verified step by step guidance

Step 1: Recognize that the equation involves a logarithm and an algebraic expression. The equation is \(3x - 15 = \log_x 1\).

Step 2: Recall that \(\log_x 1 = 0\) for any base \(x > 0\) and \(x \neq 1\), because any number raised to the power of 0 is 1.

Step 3: Substitute \(\log_x 1 = 0\) into the equation, simplifying it to \(3x - 15 = 0\).

Step 4: Solve the linear equation \(3x - 15 = 0\) by adding 15 to both sides to get \(3x = 15\).

Step 5: Divide both sides by 3 to isolate \(x\), resulting in \(x = 5\).

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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. The equation log_b(a) = c means that b^c = a. In this context, understanding how to manipulate logarithmic expressions is crucial for solving equations involving logs, especially when combined with algebraic terms.

Properties of Logarithms

Properties of logarithms, such as the product, quotient, and power rules, allow us to simplify and solve logarithmic equations. For instance, log_b(mn) = log_b(m) + log_b(n) and log_b(m/n) = log_b(m) - log_b(n) are essential for breaking down complex logarithmic expressions into manageable parts.

Solving Linear Equations

Solving linear equations involves isolating the variable to find its value. In the equation 3x - 15 = log_x(1), we need to manipulate the linear part (3x - 15) to equate it to the logarithmic expression, which may require understanding the conditions under which logarithms are defined, such as x > 0 and x ≠ 1.