Precalculus Examples

$6 \ln (e) + 4 \ln (1) - 9 \ln (3 \sqrt{e}) + \ln (e^{4})$

Step 1

Simplify each term.

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Step 1.1

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

$6 \cdot 1 + 4 \ln (1) - 9 \ln (3 \sqrt{e}) + \ln (e^{4})$

Step 1.2

Multiply $6$ by $1$ .

$6 + 4 \ln (1) - 9 \ln (3 \sqrt{e}) + \ln (e^{4})$

Step 1.3

The natural logarithm of $1$ is $0$ .

$6 + 4 \cdot 0 - 9 \ln (3 \sqrt{e}) + \ln (e^{4})$

Step 1.4

Multiply $4$ by $0$ .

$6 + 0 - 9 \ln (3 \sqrt{e}) + \ln (e^{4})$

Step 1.5

Simplify $- 9 \ln (3 \sqrt{e})$ by moving $9$ inside the logarithm.

$6 + 0 - \ln ({(3 \sqrt{e})}^{9}) + \ln (e^{4})$

Step 1.6

Apply the product rule to $3 \sqrt{e}$ .

$6 + 0 - \ln (3^{9} {\sqrt{e}}^{9}) + \ln (e^{4})$

Step 1.7

Raise $3$ to the power of $9$ .

$6 + 0 - \ln (19683 {\sqrt{e}}^{9}) + \ln (e^{4})$

Step 1.8

Rewrite ${\sqrt{e}}^{9}$ as $\sqrt{e^{9}}$ .

$6 + 0 - \ln (19683 \sqrt{e^{9}}) + \ln (e^{4})$

Step 1.9

Rewrite $e^{9}$ as ${(e^{4})}^{2} e$ .

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Step 1.9.1

Factor out $e^{8}$ .

$6 + 0 - \ln (19683 \sqrt{e^{8} e}) + \ln (e^{4})$

Step 1.9.2

Rewrite $e^{8}$ as ${(e^{4})}^{2}$ .

$6 + 0 - \ln (19683 \sqrt{{(e^{4})}^{2} e}) + \ln (e^{4})$

Step 1.10

Pull terms out from under the radical.

$6 + 0 - \ln (19683 (e^{4} \sqrt{e})) + \ln (e^{4})$

Step 1.11

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

$6 + 0 - \ln (19683 e^{4} \sqrt{e}) + 4 \ln (e)$

Step 1.12

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

$6 + 0 - \ln (19683 e^{4} \sqrt{e}) + 4 \cdot 1$

Step 1.13

Multiply $4$ by $1$ .

$6 + 0 - \ln (19683 e^{4} \sqrt{e}) + 4$

Step 2

Simplify by adding numbers.

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Step 2.1

Add $6$ and $0$ .

$6 - \ln (19683 e^{4} \sqrt{e}) + 4$

Step 2.2

Add $6$ and $4$ .

$10 - \ln (19683 e^{4} \sqrt{e})$

Step 3

The result can be shown in multiple forms.

Exact Form:

$10 - \ln (19683 e^{4} \sqrt{e})$

Decimal Form:

$- 4.38751059 \dots$