Problem 86 - College Algebra

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 equals the difference of the logarithms of the numerator and denominator. An example is: log base 2 of 2, calculated from log base 2 of 8 minus log base 2 of 4, equals 1.

Step 1: Describe the Quotient Rule for Logarithms

The quotient rule for logarithms states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and denominator. This can be written mathematically as: \[ \log_b(\frac{m}{n}) = \log_b(m) - \log_b(n) \] where b is the base of the logarithm, and m and n are any two positive real numbers distinct from 1.

Step 2: Give an Example

To understand this rule better, consider an example. Let \(b = 2\), \(m = 8\), and \(n = 4\). Substituting these values into the formula, the equation becomes: \[ \log_2(\frac{8}{4}) = \log_2(8) - \log_2(4) \]. On the left side, \( \frac{8}{4} = 2\), so \( \log_2(2) = 1 \). On the right side, using base 2, \( \log_2(8) = 3 \) and \( \log_2(4) = 2\). So, \( 3 - 2 = 1 \). Both sides of the equation yield the same result, proving the quotient rule for this example.