Problem 97 - Introductory and Intermediate Algebra for College Students 4th

Question

Describe the power rule for logarithms and give an example.

Step-by-Step Solution

Answer

The power rule for logarithms states that \( \log_b{a^n} = n \cdot \log_b{a} \). As for the instance \( \log_2{8^3} \), using the power rule, the solution will be 9.

Step 1: Describing the Power Rule

The power rule for logarithms states that the logarithm of a number raised to a certain power is equal to the power times the logarithm of the number itself. Mathematically, this concept is expressed as follows: \( \log_b{a^n} = n \cdot \log_b{a} \). Here, 'b' is the base of the logarithm, 'a' is the argument of the logarithm, and 'n' is the power to which 'a' is raised.

Step 2: Giving an Example

Consider an example to illustrate this rule: \( \log_2{8^3} \). According to the power rule, this can be rewritten as: \( 3 \cdot \log_2{8} \). Now, we know that \( \log_2{8} = 3 \), so substituting this we get: \( 3 \cdot 3 = 9 \)