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Chapter 12: Problem 40

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

Short Answer

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

Step by step solution

01

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.
02

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 \(\log_{4} 4^{6} = 6\).

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Most popular questions from this chapter

Without using a calculator, determine which is the greater number: \(\log _{4} 60\) or \(\log _{3} 40\)

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I estimate that \(\log _{8} 16\) lies between 1 and 2 because \(8^{1}=8\) and \(8^{2}=64\)

Explain how to use the graph of \(f(x)=2^{x}\) to obtain the \(\operatorname{graph}\) of \(g(x)=\log _{2} x\)

Graph \(f\) and \(g\) in the same viewing rectangle. Then describe the relationship of the graph of g to the graph of \(f\). $$f(x)=\ln x, g(x)=\ln (x+3)$$

Solve: \(x+3 \leq-4\) or \(2-7 x \leq 16\) (Section 9.2, Example 6)
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