Algebra Examples

$\ln (\frac{e}{\sqrt{e^{5}}})$

Step 1

Simplify the denominator.

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Step 1.1

Rewrite $e^{5}$ as ${(e^{2})}^{2} e$ .

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Step 1.1.1

Factor out $e^{4}$ .

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

Step 1.1.2

Rewrite $e^{4}$ as ${(e^{2})}^{2}$ .

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

Step 1.2

Pull terms out from under the radical.

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

Step 2

Cancel the common factor of $e$ and $e^{2}$ .

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Step 2.1

Multiply by $1$ .

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

Step 2.2

Cancel the common factors.

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Step 2.2.1

Factor $e$ out of $e^{2} \sqrt{e}$ .

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

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

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

Step 4

Combine and simplify the denominator.

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Step 4.1

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

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

Step 4.2

Move $\sqrt{e}$ .

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

Step 4.3

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

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

Step 4.4

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

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

Step 4.5

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

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

Step 4.6

Add $1$ and $1$ .

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

Step 4.7

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

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Step 4.7.1

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

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

Step 4.7.2

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

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

Step 4.7.3

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

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

Step 4.7.4

Cancel the common factor of $2$ .

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Step 4.7.4.1

Cancel the common factor.

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

Step 4.7.4.2

Rewrite the expression.

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

Step 4.7.5

Simplify.

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

Step 5

Simplify the denominator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 6

Move $e^{2}$ to the numerator using the negative exponent rule $\frac{1}{b^{n}} = b^{- n}$ .

$\ln (\sqrt{e} e^{- 2})$

Step 7

Rewrite $\ln (\sqrt{e} e^{- 2})$ as $\ln (\sqrt{e}) + \ln (e^{- 2})$ .

$\ln (\sqrt{e}) + \ln (e^{- 2})$

Step 8

Use logarithm rules to move $- 2$ out of the exponent.

$\ln (\sqrt{e}) - 2 \ln (e)$

Step 9

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

$\ln (\sqrt{e}) - 2 \cdot 1$

Step 10

Multiply $- 2$ by $1$ .

$\ln (\sqrt{e}) - 2$

Step 11

The result can be shown in multiple forms.

Exact Form:

$\ln (\sqrt{e}) - 2$

Decimal Form:

$- 1.5$