Evaluate log base 11 of 1/( square root of 11)

$\log_{11} (\frac{1}{\sqrt{11}})$

Multiply $\frac{1}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\log_{11} (\frac{1}{\sqrt{11}} \cdot \frac{\sqrt{11}}{\sqrt{11}})$

Combine and simplify the denominator.

Multiply $\frac{1}{\sqrt{11}}$ by $\frac{\sqrt{11}}{\sqrt{11}}$ .

$\log_{11} (\frac{\sqrt{11}}{\sqrt{11} \sqrt{11}})$

Raise $\sqrt{11}$ to the power of $1$ .

$\log_{11} (\frac{\sqrt{11}}{{\sqrt{11}}^{1} \sqrt{11}})$

Raise $\sqrt{11}$ to the power of $1$ .

$\log_{11} (\frac{\sqrt{11}}{{\sqrt{11}}^{1} {\sqrt{11}}^{1}})$

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

$\log_{11} (\frac{\sqrt{11}}{{\sqrt{11}}^{1 + 1}})$

Add $1$ and $1$ .

$\log_{11} (\frac{\sqrt{11}}{{\sqrt{11}}^{2}})$

Rewrite ${\sqrt{11}}^{2}$ as $11$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{11}$ as $11^{\frac{1}{2}}$ .

$\log_{11} (\frac{\sqrt{11}}{{(11^{\frac{1}{2}})}^{2}})$

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

$\log_{11} (\frac{\sqrt{11}}{11^{\frac{1}{2} \cdot 2}})$

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

$\log_{11} (\frac{\sqrt{11}}{11^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log_{11} (\frac{\sqrt{11}}{11^{\frac{2}{2}}})$

Rewrite the expression.

$\log_{11} (\frac{\sqrt{11}}{11^{1}})$

Evaluate the exponent.

$\log_{11} (\frac{\sqrt{11}}{11})$

Rewrite $\frac{\sqrt{11}}{11}$ as $\sqrt{11} \cdot 11^{- 1}$ .

$\log_{11} (\sqrt{11} \cdot 11^{- 1})$

Rewrite $\log_{11} (\sqrt{11} \cdot 11^{- 1})$ as $\log_{11} (\sqrt{11}) + \log_{11} (11^{- 1})$ .

$\log_{11} (\sqrt{11}) + \log_{11} (11^{- 1})$

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

$\log_{11} (\sqrt{11}) - \log_{11} (11)$

Logarithm base $11$ of $11$ is $1$ .

$\log_{11} (\sqrt{11}) - 1 \cdot 1$

Multiply $- 1$ by $1$ .

$\log_{11} (\sqrt{11}) - 1$

Rewrite $\log_{11} (\sqrt{11})$ 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 $b = 10$ .

$\frac{\log (\sqrt{11})}{\log (11)} - 1$

The result can be shown in multiple forms.

Exact Form:

$\frac{\log (\sqrt{11})}{\log (11)} - 1$

Decimal Form:

$- 0.5$

Evaluate log base 11 of 1/( square root of 11)

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