Problem 25
Question
Evaluate each expression without using a calculator. $$\log _{7} \sqrt{7}$$
Step-by-Step Solution
Verified Answer
The solution to \(\log_7 \sqrt{7}\) is 0.5.
1Step 1: Express Square Root as an Exponent
Rewrite the square root of 7 as pow(7, 0.5) to then bring the exponent down in front of the logarithm with another property: \(\log_b(a^n) = n \cdot \log_b a\). This results in: \(\log_7 (7^{0.5})\).
2Step 2: Apply the Exponent Property of Logarithms
Using the property where the exponent comes out in front, the logarithm simplifies to: \(0.5 \cdot \log_7 7\).
3Step 3: Simplify Logarithmic Term
The term \(\log_7 7\) simplifies to 1 as by definition, the logarithm of a number to its own base equals 1, resulting in: \(0.5 \cdot 1\).
4Step 4: Final Calculation
After multiplying 0.5 by 1, the final answer is 0.5.
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