Problem 34
Question
Evaluate each expression without using a calculator. $$\log _{81} 9$$
Step-by-Step Solution
Verified Answer
The result of evaluating \(\log _{81} 9\) is 8.
1Step 1: 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: \(\log_{3^4}(3^2)\)
2Step 2: 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: \(4 \log_{3}(3^2)\)
3Step 3: 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: \(4 \times 2\)
4Step 4: Solve the operation
Multiply the values which results in the final answer: 8
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