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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1:17 minutes

Problem 15

Textbook Question

In Exercises 1–40, use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator.logb x^3

Verified step by step guidance

Recognize that the expression is a logarithm of a power: \( \log_b (x^3) \).

Apply the power rule of logarithms, which states that \( \log_b (x^n) = n \cdot \log_b (x) \).

Using the power rule, rewrite the expression as \( 3 \cdot \log_b (x) \).

The expression is now expanded as much as possible using the properties of logarithms.

If \( b \) and \( x \) are known, evaluate \( \log_b (x) \) to simplify further.

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Key Concepts

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

Properties of Logarithms

The properties of logarithms are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule (logb(mn) = logb(m) + logb(n)), the quotient rule (logb(m/n) = logb(m) - logb(n)), and the power rule (logb(m^n) = n * logb(m)). Understanding these properties is essential for expanding and simplifying logarithmic expressions.

Power Rule of Logarithms

The power rule of logarithms states that the logarithm of a number raised to an exponent can be expressed as the exponent multiplied by the logarithm of the base number. For example, logb(x^3) can be rewritten as 3 * logb(x). This rule is particularly useful for expanding logarithmic expressions involving powers.

Logarithmic Evaluation

Evaluating logarithmic expressions involves finding the value of the logarithm based on known values or properties. For instance, if the base and the argument of the logarithm are known, one can directly compute the logarithm. In cases where the logarithm cannot be easily evaluated, properties of logarithms can help simplify the expression for easier computation.