Problem 106 - College Algebra Essentials

Question

Describe the quotient rule for logarithms and give an example.

Step-by-Step Solution

Answer

The quotient rule for logarithms states that the logarithm of a quotient of two numbers equals the difference of the logarithms of these two numbers. For instance, \( \log-b(\frac{m}{n}) = \log_b m - \log_b n \). In the example of \( \log_2(8/4) \), using the quotient rule gives the result of 1.

Step 1: Description of the Quotient Rule

The quotient rule for logarithms states that the logarithm of a quotient of two numbers is the difference of the logarithms of those two numbers. In other words, if you have \( \log_b(\frac{m}{n}) \), it can be written as \( \log_b m - \log_b n \), where b, m, and n are real numbers, and both b and \( \frac{m}{n} \) are greater than zero.

Step 2: Writing the Quotient Rule Formula

The quotient rule of logarithms can be expressed mathematically as follows: \[ \log_b(\frac{m}{n}) = \log_b m - \log_b n \] This is the formal way of expressing the rule.

Step 3: Providing an Example

For example, if you have \( \log_2(8/4) \) (log base 2 of 8 divided by 4). Using the quotient rule, you can express this as \( \log_2 8 - \log_2 4 \). So the calculation would then be: \( 3-2 =1\)