Precalculus Examples

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

Step 1

Rewrite $\frac{x}{e^{4}}$ as $x {(e^{4})}^{- 1}$ .

$\ln (x {(e^{4})}^{- 1}) \ln (x e^{4})$

Step 2

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

$\ln (x e^{4 \cdot - 1}) \ln (x e^{4})$

Step 3

Rewrite $\ln (x e^{4 \cdot - 1})$ as $\ln (x) + \ln (e^{4 \cdot - 1})$ .

$(\ln (x) + \ln (e^{4 \cdot - 1})) \ln (x e^{4})$

Step 4

Use logarithm rules to move $4 \cdot - 1$ out of the exponent.

$(\ln (x) + 4 \cdot - 1 \ln (e)) \ln (x e^{4})$

Step 5

Multiply $4$ by $- 1$ .

$(\ln (x) - 4 \ln (e)) \ln (x e^{4})$

Step 6

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

$(\ln (x) - 4 \cdot 1) \ln (x e^{4})$

Step 7

Multiply $- 4$ by $1$ .

$(\ln (x) - 4) \ln (x e^{4})$

Step 8

Rewrite $\ln (x e^{4})$ as $\ln (x) + \ln (e^{4})$ .

$(\ln (x) - 4) (\ln (x) + \ln (e^{4}))$

Step 9

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

$(\ln (x) - 4) (\ln (x) + 4 \ln (e))$

Step 10

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

$(\ln (x) - 4) (\ln (x) + 4 \cdot 1)$

Step 11

Multiply $4$ by $1$ .

$(\ln (x) - 4) (\ln (x) + 4)$

Step 12

Expand $(\ln (x) - 4) (\ln (x) + 4)$ using the FOIL Method.

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Step 12.1

Apply the distributive property.

$\ln (x) (\ln (x) + 4) - 4 (\ln (x) + 4)$

Step 12.2

Apply the distributive property.

$\ln (x) \ln (x) + \ln (x) \cdot 4 - 4 (\ln (x) + 4)$

Step 12.3

Apply the distributive property.

$\ln (x) \ln (x) + \ln (x) \cdot 4 - 4 \ln (x) - 4 \cdot 4$

Step 13

Simplify and combine like terms.

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Step 13.1

Simplify each term.

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Step 13.1.1

Multiply $\ln (x) \ln (x)$ .

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Step 13.1.1.1

Raise $\ln (x)$ to the power of $1$ .

$\ln^{1} (x) \ln (x) + \ln (x) \cdot 4 - 4 \ln (x) - 4 \cdot 4$

Step 13.1.1.2

Raise $\ln (x)$ to the power of $1$ .

$\ln^{1} (x) \ln^{1} (x) + \ln (x) \cdot 4 - 4 \ln (x) - 4 \cdot 4$

Step 13.1.1.3

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

${\ln (x)}^{1 + 1} + \ln (x) \cdot 4 - 4 \ln (x) - 4 \cdot 4$

Step 13.1.1.4

Add $1$ and $1$ .

$\ln^{2} (x) + \ln (x) \cdot 4 - 4 \ln (x) - 4 \cdot 4$

Step 13.1.2

Move $4$ to the left of $\ln (x)$ .

$\ln^{2} (x) + 4 \cdot \ln (x) - 4 \ln (x) - 4 \cdot 4$

Step 13.1.3

Multiply $- 4$ by $4$ .

$\ln^{2} (x) + 4 \ln (x) - 4 \ln (x) - 16$

Step 13.2

Subtract $4 \ln (x)$ from $4 \ln (x)$ .

$\ln^{2} (x) + 0 - 16$

Step 13.3

Add $\ln^{2} (x)$ and $0$ .

$\ln^{2} (x) - 16$