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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1:09 minutes

Problem 17

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.log N^(-6)

Verified step by step guidance

Recognize that the expression is a logarithm of a power: \( \log(N^{-6}) \).

Use the power rule of logarithms, which states \( \log_b(x^a) = a \cdot \log_b(x) \), to bring the exponent in front of the logarithm.

Apply the power rule: \( \log(N^{-6}) = -6 \cdot \log(N) \).

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

If \( N \) is a specific number, you could evaluate \( \log(N) \) if possible, but without a specific value, this is the expanded form.

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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 (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^p) = p * log_b(M)). Understanding these properties is essential for expanding and simplifying logarithmic expressions.

Negative Exponents

Negative exponents indicate the reciprocal of the base raised to the positive exponent. For example, N^(-6) can be rewritten as 1/(N^6). This concept is crucial when dealing with logarithmic expressions involving negative exponents, as it allows for the application of logarithmic properties effectively.

Logarithmic Evaluation

Evaluating logarithmic expressions involves determining the value of the logarithm based on its definition. For instance, log_b(a) answers the question: 'To what power must b be raised to obtain a?' In the context of expanding log N^(-6), understanding how to evaluate logarithms without a calculator can help in simplifying the expression further.