Problem 85 - 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 \( \log_{b}(mn) = \log_{10}(m) + \log_{10}(n) \). An example is \(\log_{10}(20)\) which equals \( \log_{10}(2) + \log_{10}(10) \), and approximately equals 1.3010.

Step 1: Definition of the Product Rule for Logarithms

The product rule of logarithms states that the logarithm of a product of numbers is the sum of the logarithms of all those numbers. Mathematically, this rule can be written as \( \log_{b}(mn) = \log_{b}(m) + \log_{b}(n) \) where \( b \) is the base and \( m \) and \( n \) are the numbers.

Step 2: Example illustrating the Product Rule

Let's take an example: suppose we want to find the value of \( \log_{10}(20) \). Using the product rule for logarithms we can break it down as \( \log_{10}(2*10) \). If we apply the product rule now, we get: \( \log_{10}(2) + \log_{10}(10) \)

Step 3: Calculating the individual Logarithms

Now, we can calculate each of these logarithms separately. Since log base 10 of 10 equals 1, our previous step reduce to \( \log_{10}(2) + 1 \). Now, using the value of \( \log_{10}(2) \) as approximately 0.3010, the previous equation becomes \( 0.3010 + 1 = 1.3010 \)