Precalculus Examples

$\ln (\frac{{(9 x)}^{2}}{\sqrt[4]{y^{3}}})$

Step 1

Rewrite $\ln (\frac{{(9 x)}^{2}}{\sqrt[4]{y^{3}}})$ as $\ln ({(9 x)}^{2}) - \ln (\sqrt[4]{y^{3}})$ .

$\ln ({(9 x)}^{2}) - \ln (\sqrt[4]{y^{3}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{y^{3}}$ as $y^{\frac{3}{4}}$ .

$\ln ({(9 x)}^{2}) - \ln (y^{\frac{3}{4}})$

Step 3

Expand $\ln ({(9 x)}^{2})$ by moving $2$ outside the logarithm.

$2 \ln (9 x) - \ln (y^{\frac{3}{4}})$

Step 4

Expand $\ln (y^{\frac{3}{4}})$ by moving $\frac{3}{4}$ outside the logarithm.

$2 \ln (9 x) - (\frac{3}{4} \ln (y))$

Step 5

Combine $\frac{3}{4}$ and $\ln (y)$ .

$2 \ln (9 x) - \frac{3 \ln (y)}{4}$

Step 6

Rewrite $\ln (9 x)$ as $\ln (9) + \ln (x)$ .

$2 (\ln (9) + \ln (x)) - \frac{3 \ln (y)}{4}$

Step 7

Simplify each term.

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Step 7.1

Simplify each term.

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Step 7.1.1

Rewrite $\ln (9)$ as $\ln (3^{2})$ .

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

Step 7.1.2

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

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Multiply $2$ by $2$ .

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