Precalculus Examples

$6 \ln (8 x) - \frac{1}{2} \cdot \ln (y - 3)$

Step 1

Rewrite $\ln (8 x)$ as $\ln (8) + \ln (x)$ .

$6 (\ln (8) + \ln (x)) - \frac{1}{2} \cdot \ln (y - 3)$

Step 2

Simplify each term.

Tap for more steps...

Step 2.1

Simplify each term.

Tap for more steps...

Step 2.1.1

Rewrite $\ln (8)$ as $\ln (2^{3})$ .

$6 (\ln (2^{3}) + \ln (x)) - \frac{1}{2} \cdot \ln (y - 3)$

Step 2.1.2

Expand $\ln (2^{3})$ by moving $3$ outside the logarithm.

$6 (3 \ln (2) + \ln (x)) - \frac{1}{2} \cdot \ln (y - 3)$

Step 2.2

Apply the distributive property.

$6 (3 \ln (2)) + 6 \ln (x) - \frac{1}{2} \cdot \ln (y - 3)$

Step 2.3

Multiply $3$ by $6$ .

$18 \ln (2) + 6 \ln (x) - \frac{1}{2} \cdot \ln (y - 3)$

Step 2.4

Combine $\ln (y - 3)$ and $\frac{1}{2}$ .

$18 \ln (2) + 6 \ln (x) - \frac{\ln (y - 3)}{2}$

Step 3

To write $18 \ln (2)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$6 \ln (x) + 18 \ln (2) \cdot \frac{2}{2} - \frac{\ln (y - 3)}{2}$

Step 4

Combine $18 \ln (2)$ and $\frac{2}{2}$ .

$6 \ln (x) + \frac{18 \ln (2) \cdot 2}{2} - \frac{\ln (y - 3)}{2}$

Step 5

Combine the numerators over the common denominator.

$6 \ln (x) + \frac{18 \ln (2) \cdot 2 - \ln (y - 3)}{2}$

Step 6

Multiply $2$ by $18$ .

$6 \ln (x) + \frac{36 \ln (2) - \ln (y - 3)}{2}$