Precalculus Examples

$\ln (\frac{e^{4}}{\sqrt[5]{e}})$

Step 1

Multiply $\frac{e^{4}}{\sqrt[5]{e}}$ by $\frac{{\sqrt[5]{e}}^{4}}{{\sqrt[5]{e}}^{4}}$ .

$\ln (\frac{e^{4}}{\sqrt[5]{e}} \cdot \frac{{\sqrt[5]{e}}^{4}}{{\sqrt[5]{e}}^{4}})$

Step 2

Combine and simplify the denominator.

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Step 2.1

Multiply $\frac{e^{4}}{\sqrt[5]{e}}$ by $\frac{{\sqrt[5]{e}}^{4}}{{\sqrt[5]{e}}^{4}}$ .

$\ln (\frac{e^{4} {\sqrt[5]{e}}^{4}}{\sqrt[5]{e} {\sqrt[5]{e}}^{4}})$

Step 2.2

Raise $\sqrt[5]{e}$ to the power of $1$ .

$\ln (\frac{e^{4} {\sqrt[5]{e}}^{4}}{{\sqrt[5]{e}}^{1} {\sqrt[5]{e}}^{4}})$

Step 2.3

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

$\ln (\frac{e^{4} {\sqrt[5]{e}}^{4}}{{\sqrt[5]{e}}^{1 + 4}})$

Step 2.4

Add $1$ and $4$ .

$\ln (\frac{e^{4} {\sqrt[5]{e}}^{4}}{{\sqrt[5]{e}}^{5}})$

Step 2.5

Rewrite ${\sqrt[5]{e}}^{5}$ as $e$ .

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Step 2.5.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{e}$ as $e^{\frac{1}{5}}$ .

$\ln (\frac{e^{4} {\sqrt[5]{e}}^{4}}{{(e^{\frac{1}{5}})}^{5}})$

Step 2.5.2

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

$\ln (\frac{e^{4} {\sqrt[5]{e}}^{4}}{e^{\frac{1}{5} \cdot 5}})$

Step 2.5.3

Combine $\frac{1}{5}$ and $5$ .

$\ln (\frac{e^{4} {\sqrt[5]{e}}^{4}}{e^{\frac{5}{5}}})$

Step 2.5.4

Cancel the common factor of $5$ .

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Step 2.5.4.1

Cancel the common factor.

$\ln (\frac{e^{4} {\sqrt[5]{e}}^{4}}{e^{\frac{5}{5}}})$

Step 2.5.4.2

Rewrite the expression.

$\ln (\frac{e^{4} {\sqrt[5]{e}}^{4}}{e^{1}})$

Step 2.5.5

Simplify.

$\ln (\frac{e^{4} {\sqrt[5]{e}}^{4}}{e})$

Step 3

Cancel the common factor of $e^{4}$ and $e$ .

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Step 3.1

Factor $e$ out of $e^{4} {\sqrt[5]{e}}^{4}$ .

$\ln (\frac{e (e^{3} {\sqrt[5]{e}}^{4})}{e})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Multiply by $1$ .

$\ln (\frac{e (e^{3} {\sqrt[5]{e}}^{4})}{e \cdot 1})$

Step 3.2.2

Cancel the common factor.

$\ln (\frac{e (e^{3} {\sqrt[5]{e}}^{4})}{e \cdot 1})$

Step 3.2.3

Rewrite the expression.

$\ln (\frac{e^{3} {\sqrt[5]{e}}^{4}}{1})$

Step 3.2.4

Divide $e^{3} {\sqrt[5]{e}}^{4}$ by $1$ .

$\ln (e^{3} {\sqrt[5]{e}}^{4})$

Step 4

Rewrite $\ln (e^{3} {\sqrt[5]{e}}^{4})$ as $\ln (e^{3}) + \ln ({\sqrt[5]{e}}^{4})$ .

$\ln (e^{3}) + \ln ({\sqrt[5]{e}}^{4})$

Step 5

Use logarithm rules to move $3$ out of the exponent.

$3 \ln (e) + \ln ({\sqrt[5]{e}}^{4})$

Step 6

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

$3 \cdot 1 + \ln ({\sqrt[5]{e}}^{4})$

Step 7

Multiply $3$ by $1$ .

$3 + \ln ({\sqrt[5]{e}}^{4})$

Step 8

Expand $\ln ({\sqrt[5]{e}}^{4})$ by moving $4$ outside the logarithm.

$3 + 4 \ln (\sqrt[5]{e})$

Step 9

The result can be shown in multiple forms.

Exact Form:

$3 + 4 \ln (\sqrt[5]{e})$

Decimal Form:

$3.8$