Precalculus Examples

$\ln (\frac{4 x^{3}}{\sqrt{y} {(x - 2)}^{6}})$

Step 1

Rewrite $\ln (\frac{4 x^{3}}{\sqrt{y} {(x - 2)}^{6}})$ as $\ln (4 x^{3}) - \ln (\sqrt{y} {(x - 2)}^{6})$ .

$\ln (4 x^{3}) - \ln (\sqrt{y} {(x - 2)}^{6})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{y}$ as $y^{\frac{1}{2}}$ .

$\ln (4 x^{3}) - \ln (y^{\frac{1}{2}} {(x - 2)}^{6})$

Step 3

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

$\ln (4) + \ln (x^{3}) - \ln (y^{\frac{1}{2}} {(x - 2)}^{6})$

Step 4

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

$\ln (4) + 3 \ln (x) - \ln (y^{\frac{1}{2}} {(x - 2)}^{6})$

Step 5

Rewrite $\ln (y^{\frac{1}{2}} {(x - 2)}^{6})$ as $\ln (y^{\frac{1}{2}}) + \ln ({(x - 2)}^{6})$ .

$\ln (4) + 3 \ln (x) - (\ln (y^{\frac{1}{2}}) + \ln ({(x - 2)}^{6}))$

Step 6

Expand $\ln (y^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\ln (4) + 3 \ln (x) - (\frac{1}{2} \ln (y) + \ln ({(x - 2)}^{6}))$

Step 7

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

$\ln (4) + 3 \ln (x) - (\frac{1}{2} \ln (y) + 6 \ln (x - 2))$

Step 8

Simplify each term.

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Step 8.1

Rewrite $\ln (4)$ as $\ln (2^{2})$ .

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

To write $6 \ln (x - 2)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.5

Combine $6 \ln (x - 2)$ and $\frac{2}{2}$ .

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

Step 8.6

Combine the numerators over the common denominator.

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

Step 8.7

Multiply $2$ by $6$ .

$2 \ln (2) + 3 \ln (x) - \frac{\ln (y) + 12 \ln (x - 2)}{2}$