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Chapter 12: Problem 77

Evaluate each expression without using a calculator. $$\log _{3}\left(\log _{7} 7\right)$$

Short Answer

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Step by step solution

01

Evaluate the Inner Logarithm

The first task is to simplify the expression \(\log _{7} 7\). Because the base of the logarithm and the number are the same (both 7), we can apply the property of logarithms that \(\log_b b = 1\). Thus, \(\log _{7} 7 = 1\).
02

Evaluate the Outer Logarithm

Now, we evaluate the outer logarithm using the result from Step 1. The expression \(\log _{3}(1)\) is the logarithm base 3 of 1. Since any number to the power of 0 is 1, we have that \(\log _{3}(1) = 0\).

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Most popular questions from this chapter

Solve each equation. $$3^{x^{2}-12}=9^{2 x}$$

Evaluate each expression without using a calculator. $$\log _{2}\left(\log _{3} 81\right)$$

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

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When graphing a logarithmic function, I like to show the graph of its horizontal asymptote.

Describe the change-of-base property and give an example.
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