Precalculus Examples

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[4]{x^{2}}$ as $x^{\frac{2}{4}}$ .

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

Step 3

Cancel the common factor of $2$ and $4$ .

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Step 3.1

Factor $2$ out of $2$ .

$\ln (e^{3} x^{\frac{2 (1)}{4}}) - \ln (\sqrt[3]{y})$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Factor $2$ out of $4$ .

$\ln (e^{3} x^{\frac{2 \cdot 1}{2 \cdot 2}}) - \ln (\sqrt[3]{y})$

Step 3.2.2

Cancel the common factor.

$\ln (e^{3} x^{\frac{2 \cdot 1}{2 \cdot 2}}) - \ln (\sqrt[3]{y})$

Step 3.2.3

Rewrite the expression.

$\ln (e^{3} x^{\frac{1}{2}}) - \ln (\sqrt[3]{y})$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{y}$ as $y^{\frac{1}{3}}$ .

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

Step 5

Expand $\ln (y^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

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

Step 6

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

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

Step 7

Rewrite $\ln (e^{3} x^{\frac{1}{2}})$ as $\ln (e^{3}) + \ln (x^{\frac{1}{2}})$ .

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

Step 8

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

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

Step 9

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

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

Step 10

The natural logarithm of $e$ is $1$ .

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

Step 11

Multiply $3$ by $1$ .

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

Step 12

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

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