Problem 99 - Introductory and Intermediate Algebra for College Students 4th

Question

Describe the change-of-base property and give an example.

Step-by-Step Solution

Answer

The Change-of-Base property in logarithms allows logarithm base alterations. It is represented as \(\log_c b = \frac{\log_a b}{\log_a c}\). An example is \(\log_4 8 = \frac{\log_2 8}{\log_2 4} = 1.5\).

Step 1: Defining the Change-of-Base Property

The Change-of-Base property states that for any positive numbers \(a\), \(b\), and \(c\) where \(a \neq 1\) and \(b \neq 1\), the quotient of the logarithm of \(b\) with base \(a\) and the logarithm of \(c\) with base \(a\) is equal to the logarithm of \(b\) with base \(c\). Mathematically, it can be stated as: \[\log_c b = \frac{\log_a b}{\log_a c}\]

Step 2: Giving an Example

Let's illustrate the Change-of-Base property with an example. Consider \(a = 2\), \(b = 8\), and \(c = 4\). The aim is to calculate \( \log_4 8 \). Utilizing the Change-of-Base property: \[ \log_4 8 = \frac{\log_2 8}{\log_2 4} = \frac{3}{2} = 1.5 \]