Problem 109 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The change-of-base property for logarithms states that \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\), meaning it is possible to change the base of a logarithm by dividing the logarithm of the number with the new base by the logarithm of the old base with the new base. An example of this property is the conversion of \(\log_2(8)\) into a base 10 logarithm, resulting in \(\frac{\log_{10}(8)}{\log_{10}(2)} = 3\), which matches the original value.

Step 1: Understanding the change-of-base property

The change-of-base property is a property of logarithms that indicates how to convert a logarithm of one base to a logarithm of another base. The property can be expressed as follows: Given a logarithm \(\log_b(a)\), you can change the base \(b\) to a new base \(c\) using the formula \(\log_b(a) = \frac{\log_c(a)}{\log_c(b)}\). In other words, the logarithm base \(b\) of a number \(a\) is equal to the logarithm base \(c\) of \(a\) divided by the logarithm base \(c\) of \(b\).

Step 2: Providing an example of the change-of-base property

Let's consider \(\log_2(8)\) as an example, which means 'To what power do we need to raise 2 to get 8?' The answer is 3, since \(2^3 = 8\). Now, using the change-of-base property, let's convert this base 2 logarithm to a base 10 logarithm. By the change-of-base property, \(\log_2(8)\) equals \(\frac{\log_{10}(8)}{\log_{10}(2)}\). Calculating these values, we get approximately \(\frac{0.9031}{0.301} = 3\), which confirms our earlier result.