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Chapter 12: Problem 34

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

Short Answer

Expert verified
The result of evaluating \(\log _{81} 9\) is 8.

Step by step solution

01

Set common base.

Express 81 and 9 with the same base. Recall that \(81 = 3^4\) and \(9 = 3^2\). So the expression can be written as: \(\log_{3^4}(3^2)\)
02

Apply the power rule of logarithm.

Use the power rule for logarithms which is \(\log_b(a^n) = n \log_b(a)\). This allows us to take the exponent of the base outside the logarithm: \(4 \log_{3}(3^2)\)
03

Cancel terms.

The term \(\log_{3}(3^2)\) reduces to 2 since the logarithm base is the same as the number inside the logarithm. This leaves us with the final reduced expression: \(4 \times 2\)
04

Solve the operation

Multiply the values which results in the final answer: 8

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Most popular questions from this chapter

Graph \(y=\log x, y=\log (10 x),\) and \(y=\log (0.1 x)\) in the same viewing rectangle. Describe the relationship among the three graphs. What logarithmic property accounts for this relationship?

a. Evaluate: \(\log _{2} 16\) b. Evaluate: \(\log _{2} 32-\log _{2} 2\) c. What can you conclude about $$\log _{2} 16, \text { or } \log _{2}\left(\frac{32}{2}\right) ?$$

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 \(8^{2}=64\)

Describe the power rule for logarithms and give an example.

Write as a single term that does not contain a logarithm: $$e^{\ln 8 x^{5}-\ln 2 x^{2}}$$
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