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Chapter 4: Problem 34

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

Short Answer

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

Step by step solution

01

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.
02

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.
03

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.

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