Problem 109 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The change-of-base property states that we can rewrite a logarithm in one base in terms of a different base. An example is \(log_5 100\), which can be rewritten as \(\frac{log_{10}100 }{ log_{10}5 }\) or approximately 2.86 when calculated.

Step 1: Describe the Change-of-Base Property

The change-of-base rule for logarithms states that for any positive number \(a\), \(b\) and \(c\), where \(a ≠ 1\) and \(b ≠ 1\), the change of base formula is: \(log_b{a} = \frac{log_c{a}}{log_c{b}}\). This formula allows a logarithm in one base to be rewritten in terms of a different base.

Step 2: Providing an Example

To illustrate this concept, let's consider an example: Suppose we want to change the base of \(log_5 100\). Using the change of base formula, we can write it as: \(log_5 100 = \frac{log_{10}100 }{ log_{10}5 }\). By calculating, we get \(log_5 100 = \frac{2}{0.699} ≈ 2.86\)