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Chapter 3: Problem 105

Describe the product rule for logarithms and give an example.

Short Answer

Expert verified
The product rule for logarithms states that the logarithm of a product equals the sum of the logarithms of the factors. For example, for base 2 and the numbers 4 and 8, both \( \log_2 (4*8) \) and \( \log_2 4 + \log_2 8 \) result in 5.

Step by step solution

01

Description of the Product Rule for Logarithms

The product rule for logarithms states that the logarithm of a product is equal to the sum of the logarithms of its factors. Mathematically, this can be represented as follows: if \( b \), \( m \), and \( n \) are positive real numbers with \( b 鈮� 1 \), and \( m \) and \( n \) are both positive, then \( \log_b (mn) = \log_b m + \log_b n \). In words, the logarithm base \( b \) of the product \( mn \) is equal to the sum of the logarithm base \( b \) of \( m \) and the logarithm base \( b \) of \( n \).
02

Example of the Product Rule for Logarithms

Let's use the number 2 as base and the numbers 4 and 8 as \( m \) and \( n \). We must verify the product rule. First, \(\log_2 (4 * 8)\) results in \( \log_2 32 \), which equals 5. Then, if we compute \(\log_2 4 + \log_2 8\) we obtain \( 2 + 3 \), which also equals 5, proving that the product rule holds in this case.

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