Calculus Examples

$\ln (\sqrt{e} \cdot e^{\frac{17}{2}})$

Step 1

Rewrite $\ln (\sqrt{e} \cdot e^{\frac{17}{2}})$ as $\ln (\sqrt{e}) + \ln (e^{\frac{17}{2}})$ .

$\ln (\sqrt{e}) + \ln (e^{\frac{17}{2}})$

Step 2

Use logarithm rules to move $\frac{17}{2}$ out of the exponent.

$\ln (\sqrt{e}) + \frac{17}{2} \ln (e)$

Step 3

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

$\ln (\sqrt{e}) + \frac{17}{2} \cdot 1$

Step 4

Multiply $\frac{17}{2}$ by $1$ .

$\ln (\sqrt{e}) + \frac{17}{2}$

Step 5

To write $\ln (\sqrt{e})$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\ln (\sqrt{e}) \cdot \frac{2}{2} + \frac{17}{2}$

Step 6

Combine fractions.

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Step 6.1

Combine $\ln (\sqrt{e})$ and $\frac{2}{2}$ .

$\frac{\ln (\sqrt{e}) \cdot 2}{2} + \frac{17}{2}$

Step 6.2

Combine the numerators over the common denominator.

$\frac{\ln (\sqrt{e}) \cdot 2 + 17}{2}$

Step 7

Move $2$ to the left of $\ln (\sqrt{e})$ .

$\frac{2 \ln (\sqrt{e}) + 17}{2}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$\frac{2 \ln (\sqrt{e}) + 17}{2}$

Decimal Form:

$9$