Problem 30 - College Algebra

Question

Evaluate each expression without using a calculator. $$\log _{81} 9$$

Step-by-Step Solution

Answer

The value of \( \log _{81} 9 \) is 2.

Step 1: Simplify the Base

First observe that 81 is the square of 9 or \( 9^2 \). So base 81 can be represented as \( 9^2 \). Therefore, \( \log _{81} 9 = \log _{9^2} 9 \)

Step 2: Apply the Property of Logarithms

We now apply the property of logarithms \( \log_b a^x = x \log_b a \). This gives us \( 2 \log_{9} 9 \)

Step 3: Evaluate the Logarithm

As \( \log_b a = x \) is equivalent to \( b^x = a \), we see that \( 9^1 = 9 \), so this determines \( \log_{9} 9 = 1 \). Hence, \( 2 \log_{9} 9 = 2 * 1 = 2 \)