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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1:48 minutes

Problem 100

Textbook Question

Work each problem. Which of the following is equivalent to ln(4x) - ln(2x) for x > 0?A. 2 ln x B. ln 2x C. (ln 4x)/(ln 2x) D. ln 2

Verified step by step guidance

Recognize that the expression ln(4x) - ln(2x) involves a property of logarithms.

Recall the logarithmic property: ln(a) - ln(b) = ln(a/b).

Apply this property to the expression: ln(4x) - ln(2x) = ln((4x)/(2x)).

Simplify the fraction inside the logarithm: (4x)/(2x) = 2.

Conclude that ln((4x)/(2x)) simplifies to ln(2).

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Key Concepts

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

Properties of Logarithms

Logarithms have specific properties that simplify expressions. One key property is the quotient rule, which states that ln(a) - ln(b) = ln(a/b). This property allows us to combine logarithmic expressions by converting subtraction into division, making it essential for solving problems involving logarithms.

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is commonly used in calculus and algebra due to its unique properties, such as the fact that the derivative of ln(x) is 1/x. Understanding how to manipulate ln expressions is crucial for solving logarithmic equations.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves applying logarithmic properties to rewrite them in a more manageable form. This includes using the product rule (ln(ab) = ln(a) + ln(b)) and the quotient rule. Mastery of these simplification techniques is vital for solving problems that require finding equivalent expressions.