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Chapter 4: 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 the product of two numbers is the sum of their logarithms. This is represented as \( \log_b(mn) = \log_b(m) + \log_b(n) \). An example is provided with \( \log_2(8 * 4) \) which simplifies to 5 using the product rule.

Step by step solution

01

Describe the product rule for logarithms

The product rule is an operation rule that deals with two numbers. The rule can be stated as: Logarithm of the product of two positive numbers equals the sum of their logarithms. Mathematically, it is written as: \( \log_b(mn) = \log_b(m) + \log_b(n) \) where \( b \), \( m \), and \( n \) are positive real numbers and \( b \) is not equal to 1.
02

Give an example

To illustrate this rule, let鈥檚 consider an example. Suppose we want to find the value of \( \log_2(8 * 4) \). According to the product rule for logarithms, we can break this down to \( \log_2(8) + \log_2(4) \). We know the \( \log_2(8) = 3 \) and \( \log_2(4) = 2 \) hence, adding these together we get 5. Therefore, \( \log_2(8 * 4) = 5 \).

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