Precalculus Examples

$(2 \ln (e^{2}) - \frac{1}{2}) (\ln (e^{4})) + \ln (e)$

Step 1

Simplify each term.

Tap for more steps...

Step 1.1

Simplify each term.

Tap for more steps...

Step 1.1.1

Use logarithm rules to move $2$ out of the exponent.

$(2 (2 \ln (e)) - \frac{1}{2}) \ln (e^{4}) + \ln (e)$

Step 1.1.2

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

$(2 (2 \cdot 1) - \frac{1}{2}) \ln (e^{4}) + \ln (e)$

Step 1.1.3

Multiply $2$ by $1$ .

$(2 \cdot 2 - \frac{1}{2}) \ln (e^{4}) + \ln (e)$

Step 1.1.4

Multiply $2$ by $2$ .

$(4 - \frac{1}{2}) \ln (e^{4}) + \ln (e)$

Step 1.2

To write $4$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$(4 \cdot \frac{2}{2} - \frac{1}{2}) \ln (e^{4}) + \ln (e)$

Step 1.3

Combine $4$ and $\frac{2}{2}$ .

$(\frac{4 \cdot 2}{2} - \frac{1}{2}) \ln (e^{4}) + \ln (e)$

Step 1.4

Combine the numerators over the common denominator.

$\frac{4 \cdot 2 - 1}{2} \ln (e^{4}) + \ln (e)$

Step 1.5

Simplify the numerator.

Tap for more steps...

Step 1.5.1

Multiply $4$ by $2$ .

$\frac{8 - 1}{2} \ln (e^{4}) + \ln (e)$

Step 1.5.2

Subtract $1$ from $8$ .

$\frac{7}{2} \ln (e^{4}) + \ln (e)$

Step 1.6

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

$\frac{7}{2} (4 \ln (e)) + \ln (e)$

Step 1.7

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

$\frac{7}{2} (4 \cdot 1) + \ln (e)$

Step 1.8

Multiply $4$ by $1$ .

$\frac{7}{2} \cdot 4 + \ln (e)$

Step 1.9

Cancel the common factor of $2$ .

Tap for more steps...

Step 1.9.1

Factor $2$ out of $4$ .

$\frac{7}{2} \cdot (2 (2)) + \ln (e)$

Step 1.9.2

Cancel the common factor.

$\frac{7}{2} \cdot (2 \cdot 2) + \ln (e)$

Step 1.9.3

Rewrite the expression.

$7 \cdot 2 + \ln (e)$

Step 1.10

Multiply $7$ by $2$ .

$14 + \ln (e)$

Step 1.11

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

$14 + 1$

Step 2

Add $14$ and $1$ .

$15$