Problem 31 - College Algebra

Question

Evaluate each expression without using a calculator. $$\log _{2} \frac{1}{\sqrt{2}}$$

Step-by-Step Solution

Answer

The value of the expression \(\log _{2} \frac{1}{\sqrt{2}}\) is \(-\frac{1}{2}\).

Step 1: Rewrite in base 2 logarithm form.

The given expression can be rewritten as: \(\log_{2}(2^{-\frac{1}{2}})\). Here, the square root of a number is the same as raising that number to the 1/2 power. Thus, \(\sqrt{2}\) can be rewritten as \(2^{\frac{1}{2}}\), and 1 divided by \(\sqrt{2}\) can be rewritten as \(2^{-\frac{1}{2}}\).

Step 2: Apply the logarithmic rule.

Using the rule of logarithms \(\log_{b}(b^{r}) = r\), where b is the base and r is the exponent (assuming that \(b > 0, b ≠ 1\), and r are any real numbers), the expression can be simplified as: \(-\frac{1}{2}\). Thus, \(\log_{2}(2^{-\frac{1}{2}}) = -\frac{1}{2}\).

Step 3: Conclusion

Hence, the value of the expression \(\log _{2} \frac{1}{\sqrt{2}}\) is \(-\frac{1}{2}\).