Problem 105 - College Algebra

Question

Describe the product rule for logarithms and give an example.

Step-by-Step Solution

Answer

The Product Rule for logarithms states that for any positive numbers M, N, and a number b (b ≠ 1), \(\log_b (MN) = \log_b M + \log_b N\). Example: \(\log_2 32 = \log_2 4 + \log_2 8 = 2 + 3 = 5\).

Step 1: Defining Logarithm Rules

The Product Rule for logarithms states that for any positive numbers \(M\), \(N\), and a number \(b\) (\(b \neq 1\)), if the base-b logarithm of \(M\) and \(N\) are known, then the base-b logarithm of the product \(MN\) can be found using the equation: \(\log_b (MN) = \log_b M + \log_b N\). This rule allows us to separate the logarithm of a product into the sum of the logarithms of its factors.

Step 2: Applying the Product Rule

Let's use an example with numbers to make the application of the rule more concrete. Suppose \(b = 2\), \(M = 4\), and \(N = 8\). Then \(\log_2 4 = 2\) and \(\log_2 8 = 3\). According to the product rule, the logarithm base 2 of the product \(4*8 = 32\) can be calculated as the sum of \(\log_2 4\) and \(\log_2 8\): \(\log_2 32 = \log_2 4 + \log_2 8 = 2 + 3 = 5\). Hence, the logarithm base 2 of 32 is 5, therefore the rule is correctly applied.