Problem 106 - 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 logarithm of the numerator minus the logarithm of the denominator. An example is \(\log_{2}{\frac{8}{4}}\), which equals \( \log_{2}{8} - \log_{2}{4} = 1\).

Step 1: Describe the Quotient Rule for Logarithms

The quotient rule for logarithms states that the logarithm of a quotient is equal to the logarithm of the numerator minus the logarithm of the denominator. Written in mathematical terms: \(\log_{b}{\frac{m}{n}} = \log_{b}{m} - \log_{b}{n}\) . Here, 'b' is the base of the logarithm, while 'm' and 'n' represent the numerator and the denominator, respectively.

Step 2: Giving an example

Here's an example illustrating the quotient rule for logarithms: Consider \(\log_{2}{\frac{8}{4}}\). According to the quotient rule, this can be rewritten as \( \log_{2}{8} - \log_{2}{4}\) .

Step 3: Calculating the Logarithm

\(\log_{2}{8} - \log_{2}{4} = 3 - 2 = 1 \). So, \(\log_{2}{\frac{8}{4}} = 1\).