Problem 105 - College Algebra Essentials

Question

Describe the product rule for logarithms and give an example.

Step-by-Step Solution

Answer

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 1: 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.

Step 2: Give an example

To illustrate this rule, let’s 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, \( \log_2(8 * 4) = 5 \).