Problem 34 - College Algebra

Question

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

Step-by-Step Solution

Answer

The solution to \( \log_{81} 9 \) is \( \frac{1}{2} \)

Step 1: Identify the Base and the Argument of the Logarithm

A logarithm has the form \( \log_b a \) where 'b' is the base of the logarithm and 'a' is the argument. In this problem, the base of the logarithm is 81 and the argument is 9.

Step 2: Understanding the logarithmic relationship

By definition, if \( \log_b a = c \) then \( b^c = a \). This means that we are looking for the power that 81 needs to be raised to in order to get 9.

Step 3: Solve for the Power

We know that 81 = 9*9 and 9 is \(3^2\). So, \(81 = (3^2)^2 = 3^{2*2}\). Therefore, \(81^{1/2} = 3^2 = 9\). So the power is 1/2.