Problem 31
Question
Evaluate each expression without using a calculator. $$\log _{2} \frac{1}{\sqrt{2}}$$
Step-by-Step Solution
Verified Answer
-0.5
1Step 1: Rewrite the fraction using exponent
Given the expression \(\log _{2} \frac{1}{\sqrt{2}}\), the fraction \(\frac{1}{\sqrt{2}}\) can be rewritten as \(2^{-0.5}\). So the expression becomes \(\log _{2}(2^{-0.5})\).
2Step 2: Apply the logarithmic properties
Using properties of logarithms \(\log_{a}(b^x) = x \cdot \log_{a}(b)\), the expression becomes \(-0.5 \cdot \log_{2}(2)\).
3Step 3: Simplify the expression
The expression \(\log_{2}(2)= 1\). Therefore, the expression becomes \(-0.5\).
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