Precalculus Examples

$\ln (\frac{1}{\sqrt{e^{7}}})$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $e^{7}$ as ${(e^{3})}^{2} e$ .

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Step 1.1.1

Factor out $e^{6}$ .

$\ln (\frac{1}{\sqrt{e^{6} e}})$

Step 1.1.2

Rewrite $e^{6}$ as ${(e^{3})}^{2}$ .

$\ln (\frac{1}{\sqrt{{(e^{3})}^{2} e}})$

Step 1.2

Pull terms out from under the radical.

$\ln (\frac{1}{e^{3} \sqrt{e}})$

Step 2

Multiply $\frac{1}{e^{3} \sqrt{e}}$ by $\frac{\sqrt{e}}{\sqrt{e}}$ .

$\ln (\frac{1}{e^{3} \sqrt{e}} \cdot \frac{\sqrt{e}}{\sqrt{e}})$

Step 3

Combine and simplify the denominator.

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Step 3.1

Multiply $\frac{1}{e^{3} \sqrt{e}}$ by $\frac{\sqrt{e}}{\sqrt{e}}$ .

$\ln (\frac{\sqrt{e}}{e^{3} \sqrt{e} \sqrt{e}})$

Step 3.2

Move $\sqrt{e}$ .

$\ln (\frac{\sqrt{e}}{e^{3} (\sqrt{e} \sqrt{e})})$

Step 3.3

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

$\ln (\frac{\sqrt{e}}{e^{3} ({\sqrt{e}}^{1} \sqrt{e})})$

Step 3.4

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

$\ln (\frac{\sqrt{e}}{e^{3} ({\sqrt{e}}^{1} {\sqrt{e}}^{1})})$

Step 3.5

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

$\ln (\frac{\sqrt{e}}{e^{3} {\sqrt{e}}^{1 + 1}})$

Step 3.6

Add $1$ and $1$ .

$\ln (\frac{\sqrt{e}}{e^{3} {\sqrt{e}}^{2}})$

Step 3.7

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

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Step 3.7.1

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

$\ln (\frac{\sqrt{e}}{e^{3} {(e^{\frac{1}{2}})}^{2}})$

Step 3.7.2

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

$\ln (\frac{\sqrt{e}}{e^{3} e^{\frac{1}{2} \cdot 2}})$

Step 3.7.3

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

$\ln (\frac{\sqrt{e}}{e^{3} e^{\frac{2}{2}}})$

Step 3.7.4

Cancel the common factor of $2$ .

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Step 3.7.4.1

Cancel the common factor.

$\ln (\frac{\sqrt{e}}{e^{3} e^{\frac{2}{2}}})$

Step 3.7.4.2

Rewrite the expression.

$\ln (\frac{\sqrt{e}}{e^{3} e^{1}})$

Step 3.7.5

Simplify.

$\ln (\frac{\sqrt{e}}{e^{3} e})$

Step 4

Multiply $e^{3}$ by $e$ by adding the exponents.

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Step 4.1

Multiply $e^{3}$ by $e$ .

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Step 4.1.1

Raise $e$ to the power of $1$ .

$\ln (\frac{\sqrt{e}}{e^{3} e^{1}})$

Step 4.1.2

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

$\ln (\frac{\sqrt{e}}{e^{3 + 1}})$

Step 4.2

Add $3$ and $1$ .

$\ln (\frac{\sqrt{e}}{e^{4}})$

Step 5

Move $e^{4}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\ln (\sqrt{e} e^{- 4})$

Step 6

Rewrite $\ln (\sqrt{e} e^{- 4})$ as $\ln (\sqrt{e}) + \ln (e^{- 4})$ .

$\ln (\sqrt{e}) + \ln (e^{- 4})$

Step 7

Use logarithm rules to move $- 4$ out of the exponent.

$\ln (\sqrt{e}) - 4 \ln (e)$

Step 8

The natural logarithm of $e$ is $1$ .

$\ln (\sqrt{e}) - 4 \cdot 1$

Step 9

Multiply $- 4$ by $1$ .

$\ln (\sqrt{e}) - 4$

Step 10

The result can be shown in multiple forms.

Exact Form:

$\ln (\sqrt{e}) - 4$

Decimal Form:

$- 3.5$