Precalculus Examples

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

Step 1

Rewrite $\ln (\sqrt{e} \cdot e^{\frac{1}{5}})$ as $\ln (\sqrt{e}) + \ln (e^{\frac{1}{5}})$ .

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

Step 2

Use logarithm rules to move $\frac{1}{5}$ out of the exponent.

$\ln (\sqrt{e}) + \frac{1}{5} \ln (e)$

Step 3

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

$\ln (\sqrt{e}) + \frac{1}{5} \cdot 1$

Step 4

Multiply $\frac{1}{5}$ by $1$ .

$\ln (\sqrt{e}) + \frac{1}{5}$

Step 5

To write $\ln (\sqrt{e})$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$\ln (\sqrt{e}) \cdot \frac{5}{5} + \frac{1}{5}$

Step 6

Combine fractions.

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Step 6.1

Combine $\ln (\sqrt{e})$ and $\frac{5}{5}$ .

$\frac{\ln (\sqrt{e}) \cdot 5}{5} + \frac{1}{5}$

Step 6.2

Combine the numerators over the common denominator.

$\frac{\ln (\sqrt{e}) \cdot 5 + 1}{5}$

Step 7

Move $5$ to the left of $\ln (\sqrt{e})$ .

$\frac{5 \ln (\sqrt{e}) + 1}{5}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{5 \ln (\sqrt{e}) + 1}{5}$

Decimal Form:

$0.7$