Problem 40 - Introductory and Intermediate Algebra for College Students 4th

Question

Evaluate each expression without using a calculator. $$\log _{4} 4^{6}$$

Step-by-Step Solution

Answer

The value of \(\log_{4} 4^{6}\) is 6.

Step 1: Understand the problem

We are given the expression \(\log_{4} 4^{6}\). Both the base of the log and the base of the power is 4.

Step 2: Applying the rules of logarithm

We know that \(\log_b b^x = x\). This is because the logarithm base \(b\) of \(b^x\) asks the question 'To what power must we raise \(b\) to get \(b^x\)?' The answer is \(x\). This matches the format of our expression here, so \(\log_{4} 4^{6} = 6\).