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Chapter 3: Problem 107

Describe the power rule for logarithms and give an example.

Short Answer

Expert verified
The power rule for logarithms states that for any base \(b\), \(\log_b (a^n) = n \cdot \log_b (a)\), where \(a\) and \(n\) are any real numbers and \(b\) is a real number not equal to 1. An example is \(\log_2 (4^3)\), which can be rewritten as \(3 \cdot \log_2 (4)\) using the power rule.

Step by step solution

01

Describe the Power Rule

The power rule for logarithms states that for any base \(b\), \(\log_b (a^n) = n \cdot \log_b (a)\), where \(a\) is a positive real number, \(b\) is a real number (not equal to 1), and \(n\) is any real number. This is useful when trying to bring down exponents in a logarithmic expression.
02

Give an Example

For example, consider the logarithm \(\log_2 (4^3)\). Using the power rule, this can be rewritten as \(3 \cdot \log_2 (4)\).

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