Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Rewrite $\ln (x^{3} y^{2})$ as $\ln (x^{3}) + \ln (y^{2})$ .

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

Step 6

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

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

Step 7

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

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

Step 8

To write $3 \ln (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

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

Step 9

Combine $3 \ln (x)$ and $\frac{2}{2}$ .

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

Step 10

Combine the numerators over the common denominator.

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

Step 11

Simplify each term.

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Step 11.1

Simplify the numerator.

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Step 11.1.1

Factor $\ln (x)$ out of $3 \ln (x) \cdot 2 - \ln (x)$ .

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Step 11.1.1.1

Factor $\ln (x)$ out of $3 \ln (x) \cdot 2$ .

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

Step 11.1.1.2

Factor $\ln (x)$ out of $- \ln (x)$ .

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

Step 11.1.1.3

Factor $\ln (x)$ out of $\ln (x) (3 \cdot 2) + \ln (x) \cdot - 1$ .

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

Step 11.1.2

Multiply $3$ by $2$ .

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

Step 11.1.3

Subtract $1$ from $6$ .

$2 \ln (y) + \frac{\ln (x) \cdot 5}{2}$

Step 11.2

Move $5$ to the left of $\ln (x)$ .

$2 \ln (y) + \frac{5 \ln (x)}{2}$