Evaluate log of 1/( square root of 1000)

$\log (\frac{1}{\sqrt{1000}})$

Simplify the denominator.

Rewrite $1000$ as $10^{2} \cdot 10$ .

Factor $100$ out of $1000$ .

$\log (\frac{1}{\sqrt{100 (10)}})$

Rewrite $100$ as $10^{2}$ .

$\log (\frac{1}{\sqrt{10^{2} \cdot 10}})$

Pull terms out from under the radical.

$\log (\frac{1}{10 \sqrt{10}})$

Multiply $\frac{1}{10 \sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\log (\frac{1}{10 \sqrt{10}} \cdot \frac{\sqrt{10}}{\sqrt{10}})$

Combine and simplify the denominator.

Multiply $\frac{1}{10 \sqrt{10}}$ by $\frac{\sqrt{10}}{\sqrt{10}}$ .

$\log (\frac{\sqrt{10}}{10 \sqrt{10} \sqrt{10}})$

Move $\sqrt{10}$ .

$\log (\frac{\sqrt{10}}{10 (\sqrt{10} \sqrt{10})})$

Raise $\sqrt{10}$ to the power of $1$ .

$\log (\frac{\sqrt{10}}{10 ({\sqrt{10}}^{1} \sqrt{10})})$

Raise $\sqrt{10}$ to the power of $1$ .

$\log (\frac{\sqrt{10}}{10 ({\sqrt{10}}^{1} {\sqrt{10}}^{1})})$

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

$\log (\frac{\sqrt{10}}{10 {\sqrt{10}}^{1 + 1}})$

Add $1$ and $1$ .

$\log (\frac{\sqrt{10}}{10 {\sqrt{10}}^{2}})$

Rewrite ${\sqrt{10}}^{2}$ as $10$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{10}$ as $10^{\frac{1}{2}}$ .

$\log (\frac{\sqrt{10}}{10 {(10^{\frac{1}{2}})}^{2}})$

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

$\log (\frac{\sqrt{10}}{10 \cdot 10^{\frac{1}{2} \cdot 2}})$

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

$\log (\frac{\sqrt{10}}{10 \cdot 10^{\frac{2}{2}}})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log (\frac{\sqrt{10}}{10 \cdot 10^{\frac{2}{2}}})$

Rewrite the expression.

$\log (\frac{\sqrt{10}}{10 \cdot 10^{1}})$

Evaluate the exponent.

$\log (\frac{\sqrt{10}}{10 \cdot 10})$

Multiply $10$ by $10$ .

$\log (\frac{\sqrt{10}}{100})$

The result can be shown in multiple forms.

Exact Form:

$\log (\frac{\sqrt{10}}{100})$

Decimal Form:

$- 1.5$