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Chapter 4: Problem 31

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

Short Answer

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

Step by step solution

01

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 \(2^{-\frac{1}{2}}\).
02

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 \(b > 0, b 鈮� 1\), and r are any real numbers), the expression can be simplified as: \(-\frac{1}{2}\). Thus, \(\log_{2}(2^{-\frac{1}{2}}) = -\frac{1}{2}\).
03

Conclusion

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

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Most popular questions from this chapter

Find the domain of each logarithmic function. $$ f(x)=\ln \left(x^{2}-x-2\right) $$

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