Question

evaluate the expression without using a calculator.
\\(\log_{2} \frac{1}{\sqrt{2}}\\)

\\(\log_{2} \frac{1}{\sqrt{2}} = \square\\)

Explanation:

Step1: Rewrite the argument

Rewrite \(\frac{1}{\sqrt{2}}\) as \(2^{-\frac{1}{2}}\) since \(\sqrt{2}=2^{\frac{1}{2}}\) and \(\frac{1}{a^n}=a^{-n}\). So the expression becomes \(\log_{2}2^{-\frac{1}{2}}\).

Step2: Apply logarithm power rule

Use the power rule of logarithms \(\log_{a}a^{x}=x\) (where \(a>0,a
eq1\)). Here \(a = 2\) and \(x=-\frac{1}{2}\), so \(\log_{2}2^{-\frac{1}{2}}=-\frac{1}{2}\).

Answer:

\(-\frac{1}{2}\)