Precalculus Examples

$e^{2} \cdot \frac{1}{2} \cdot \ln (\frac{81}{64})$

Step 1

Rewrite $\ln (\frac{81}{64})$ as $\ln (81) - \ln (64)$ .

$e^{2} \cdot \frac{1}{2} \cdot (\ln (81) - \ln (64))$

Step 2

Rewrite $\ln (64)$ as $\ln (2^{6})$ .

$e^{2} \cdot \frac{1}{2} \cdot (\ln (81) - \ln (2^{6}))$

Step 3

Expand $\ln (2^{6})$ by moving $6$ outside the logarithm.

$e^{2} \cdot \frac{1}{2} \cdot (\ln (81) - (6 \ln (2)))$

Step 4

Multiply $6$ by $- 1$ .

$e^{2} \cdot \frac{1}{2} \cdot (\ln (81) - 6 \ln (2))$

Step 5

Combine $e^{2}$ and $\frac{1}{2}$ .

$\frac{e^{2}}{2} \cdot (\ln (81) - 6 \ln (2))$

Step 6

Simplify each term.

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Step 6.1

Rewrite $\ln (81)$ as $\ln (3^{4})$ .

$\frac{e^{2}}{2} \cdot (\ln (3^{4}) - 6 \ln (2))$

Step 6.2

Expand $\ln (3^{4})$ by moving $4$ outside the logarithm.

$\frac{e^{2}}{2} \cdot (4 \ln (3) - 6 \ln (2))$

Step 7

Apply the distributive property.

$\frac{e^{2}}{2} (4 \ln (3)) + \frac{e^{2}}{2} (- 6 \ln (2))$

Step 8

Cancel the common factor of $2$ .

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Step 8.1

Factor $2$ out of $4 \ln (3)$ .

$\frac{e^{2}}{2} (2 (2 \ln (3))) + \frac{e^{2}}{2} (- 6 \ln (2))$

Step 8.2

Cancel the common factor.

$\frac{e^{2}}{2} (2 (2 \ln (3))) + \frac{e^{2}}{2} (- 6 \ln (2))$

Step 8.3

Rewrite the expression.

$e^{2} (2 \ln (3)) + \frac{e^{2}}{2} (- 6 \ln (2))$

Step 9

Cancel the common factor of $2$ .

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Step 9.1

Factor $2$ out of $- 6 \ln (2)$ .

$e^{2} (2 \ln (3)) + \frac{e^{2}}{2} (2 (- 3 \ln (2)))$

Step 9.2

Cancel the common factor.

$e^{2} (2 \ln (3)) + \frac{e^{2}}{2} (2 (- 3 \ln (2)))$

Step 9.3

Rewrite the expression.

$e^{2} (2 \ln (3)) + e^{2} (- 3 \ln (2))$

Step 10

Move $2$ to the left of $e^{2}$ .

$2 e^{2} \ln (3) + e^{2} (- 3 \ln (2))$

Step 11

Reorder factors in $2 e^{2} \ln (3) + e^{2} (- 3 \ln (2))$ .

$2 e^{2} \ln (3) - 3 e^{2} \ln (2)$