Precalculus Examples

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

Step 1

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

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

Step 2

Combine and simplify the denominator.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 2.5

Add $1$ and $1$ .

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

Step 2.6

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

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Step 2.6.1

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

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

Step 2.6.2

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

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

Step 2.6.3

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

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

Step 2.6.4

Cancel the common factor of $2$ .

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Step 2.6.4.1

Cancel the common factor.

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

Step 2.6.4.2

Rewrite the expression.

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

Step 2.6.5

Evaluate the exponent.

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

Step 3

Rewrite $\frac{\sqrt{10}}{10}$ as $\sqrt{10} \cdot 10^{- 1}$ .

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

Step 4

Rewrite $\log (\sqrt{10} \cdot 10^{- 1})$ as $\log (\sqrt{10}) + \log (10^{- 1})$ .

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

Step 5

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

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

Step 6

Logarithm base $10$ of $10$ is $1$ .

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

Step 7

Multiply $- 1$ by $1$ .

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

Step 8

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$- 0.5$