Problem 89 - College Algebra

Question

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

Step-by-Step Solution

Answer

The change-of-base property allows us to compute the logarithm of any number with respect to any base. It is represented as: \[ \log_b a = \frac{{\log_c a}}{{\log_c b}} \] where \( b, c \) and \( a \) are positive numbers and \( b, c ≠ 1 \). As an example, to calculate \( \log_2 8 \) in base 10, we apply the property and get \( \log_2 8 = \frac{{\log_{10} 8}}{{\log_{10} 2}} \), which equals to \( 3 \).

Step 1: Identifying the Change of Base Formula for Logarithms

The change of base formula is a specific method to represent the logarithm of any number in terms of any other number as a base. It's expressed as follows: \[ \log_b a = \frac{{\log_c a}}{{\log_c b}} \], where \( b, c \) and \( a \) are positive numbers and \( b, c ≠ 1 \). This formula allows changing the base of the logarithm from \( b \) to \( c \).

Step 2: Understanding the Change of Base Formula

In the equation above, \( a \) is the number we're taking the logarithm of, \( b \) is the base of the original logarithm and \( c \) is the new base. According to this formula, the log base \( b \) of \( a \) can be calculated as the log base \( c \) of \( a \) divided by the log base \( c \) of \( b \).

Step 3: Example of the Change of Base Formula

For instance, in order to calculate \( \log_2 8 \), but we only know how to compute logs with base \( 10 \), we can change the base from \( 2 \) to \( 10 \) using the change of base formula as follows: \[ \log_2 8 = \frac{{\log_{10} 8}}{{\log_{10} 2}} \]. Using calculator for the division, the answer is \( 3 \), since 2 to the power of 3 equals 8.