Precalculus Examples

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

Step 1

Rewrite $\ln (\frac{{(x - 1)}^{4} y^{3}}{\sqrt{z} (1 + x)})$ as $\ln ({(x - 1)}^{4} y^{3}) - \ln (\sqrt{z} (1 + x))$ .

$\ln ({(x - 1)}^{4} y^{3}) - \ln (\sqrt{z} (1 + x))$

Step 2

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

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

Step 3

Rewrite $\ln ({(x - 1)}^{4} y^{3})$ as $\ln ({(x - 1)}^{4}) + \ln (y^{3})$ .

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

Step 4

Expand $\ln ({(x - 1)}^{4})$ by moving $4$ outside the logarithm.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify each term.

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Step 8.1

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

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

Step 8.2

To write $\ln (1 + x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 8.3

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

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

Step 8.4

Combine the numerators over the common denominator.

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

Step 8.5

Move $2$ to the left of $\ln (1 + x)$ .

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

Step 9

To write $4 \ln (x - 1)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 10

Combine $4 \ln (x - 1)$ and $\frac{2}{2}$ .

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

Step 11

Combine the numerators over the common denominator.

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

Step 12

Simplify the numerator.

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Step 12.1

Multiply $2$ by $4$ .

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

Step 12.2

Apply the distributive property.

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

Step 12.3

Multiply $2$ by $- 1$ .

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