Problem 29
Question
Evaluate each expression without using a calculator. $$\log _{7} \sqrt{7}$$
Step-by-Step Solution
Verified Answer
Therefore, \(\log_{7}\sqrt{7} = 1/2\).
1Step 1: Convert the Root to Fractional Exponent
Represent the square root of 7 as a fractional exponent. According to the rule, the square root can be written as an exponent of 1/2. So, \( \sqrt{7} = 7^{1/2} \).
2Step 2: Apply Logarithm Property
Next, use the property of logarithms that \(log_b(a^m)=m*log_b(a)\) to break down the expression. So, we have \(log_{7}(7^{1/2})=(1/2)*log_{7}(7)\).
3Step 3: Simplify the terms
We know that \(log_{b}(b)=1\) because any number raised to power 1 gives itself. So, \(log_{7}(7)=1\). Substituting this in the equation, we get \((1/2)*log_{7}(7) = (1/2)*1 = 1/2 \).
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