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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1:18 minutes

Problem 91

Textbook Question

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement.log4 (2x^3) = 3 log4 (2x)

Verified step by step guidance

Step 1: Recall the logarithmic property that states \( \log_b (MN) = \log_b M + \log_b N \).

Step 2: Apply the property to the left side: \( \log_4 (2x^3) = \log_4 2 + \log_4 (x^3) \).

Step 3: Use the power rule of logarithms on \( \log_4 (x^3) \), which is \( 3 \log_4 x \).

Step 4: The left side becomes \( \log_4 2 + 3 \log_4 x \).

Step 5: Compare this with the right side, \( 3 \log_4 (2x) = 3(\log_4 2 + \log_4 x) = 3 \log_4 2 + 3 \log_4 x \), and determine if both sides are equal.

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Key Concepts

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

Logarithmic Properties

Logarithmic properties are rules that govern the manipulation of logarithms. Key properties include the product rule (log_b(mn) = log_b(m) + log_b(n)), the quotient rule (log_b(m/n) = log_b(m) - log_b(n)), and the power rule (log_b(m^k) = k * log_b(m)). Understanding these properties is essential for simplifying logarithmic expressions and solving equations involving logarithms.

Change of Base Formula

The change of base formula allows you to convert logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when dealing with logarithms that are not easily simplified or when using calculators that typically only compute logarithms in base 10 or e. Mastery of this formula aids in verifying the truth of logarithmic equations.

Equivalence of Logarithmic Expressions

Equivalence of logarithmic expressions involves determining whether two logarithmic statements represent the same value. This requires understanding how to manipulate and simplify logarithmic expressions using properties and rules. In the context of the given equation, verifying equivalence will involve applying logarithmic properties to both sides to see if they yield the same result, thus establishing whether the equation is true or false.