Evaluate log base 2 of 1/( square root of 2)

$\log_{2} (\frac{1}{\sqrt{2}})$

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\log_{2} (\frac{1}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}})$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{2}}$ by $\frac{\sqrt{2}}{\sqrt{2}}$ .

$\log_{2} (\frac{\sqrt{2}}{\sqrt{2} \sqrt{2}})$

Raise $\sqrt{2}$ to the power of $1$ .

$\log_{2} (\frac{\sqrt{2}}{{\sqrt{2}}^{1} \sqrt{2}})$

Raise $\sqrt{2}$ to the power of $1$ .

$\log_{2} (\frac{\sqrt{2}}{{\sqrt{2}}^{1} {\sqrt{2}}^{1}})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{2} (\frac{\sqrt{2}}{{\sqrt{2}}^{1 + 1}})$

Add $1$ and $1$ .

$\log_{2} (\frac{\sqrt{2}}{{\sqrt{2}}^{2}})$

Rewrite ${\sqrt{2}}^{2}$ as $2$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2}$ as $2^{\frac{1}{2}}$ .

$\log_{2} (\frac{\sqrt{2}}{{(2^{\frac{1}{2}})}^{2}})$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$\log_{2} (\frac{\sqrt{2}}{2^{\frac{1}{2} \cdot 2}})$

Combine $\frac{1}{2}$ and $2$ .

$\log_{2} (\frac{\sqrt{2}}{2^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log_{2} (\frac{\sqrt{2}}{2^{\frac{2}{2}}})$

Rewrite the expression.

$\log_{2} (\frac{\sqrt{2}}{2^{1}})$

Evaluate the exponent.

$\log_{2} (\frac{\sqrt{2}}{2})$

Rewrite $\frac{\sqrt{2}}{2}$ as $\sqrt{2} \cdot 2^{- 1}$ .

$\log_{2} (\sqrt{2} \cdot 2^{- 1})$

Rewrite $\log_{2} (\sqrt{2} \cdot 2^{- 1})$ as $\log_{2} (\sqrt{2}) + \log_{2} (2^{- 1})$ .

$\log_{2} (\sqrt{2}) + \log_{2} (2^{- 1})$

Use logarithm rules to move $- 1$ out of the exponent.

$\log_{2} (\sqrt{2}) - \log_{2} (2)$

Logarithm base $2$ of $2$ is $1$ .

$\log_{2} (\sqrt{2}) - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$\log_{2} (\sqrt{2}) - 1$

Rewrite $\log_{2} (\sqrt{2})$ using the change of base formula.

The change of base rule can be used if $a$ and $b$ are greater than $0$ and not equal to $1$ , and $x$ is greater than $0$ .

$\log_{a} (x) = \frac{\log_{b} (x)}{\log_{b} (a)} - 1$

Substitute in values for the variables in the change of base formula, using $b = 10$ .

$\frac{\log (\sqrt{2})}{\log (2)} - 1$

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\sqrt{2})}{\log (2)} - 1$

Decimal Form:

$- 0.5$

Evaluate log base 2 of 1/( square root of 2)

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