Calculus Examples

$\ln (\frac{x}{\sqrt{x^{4} y^{4}}})$

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{4} y^{4}}$ as ${(x^{4} y^{4})}^{\frac{1}{2}}$ .

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

Step 3

Expand $\ln ({(x^{4} y^{4})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify each term.

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Step 8.1

Cancel the common factor of $4 \ln (x) + 4 \ln (y)$ and $2$ .

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Step 8.1.1

Factor $2$ out of $4 \ln (x)$ .

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

Step 8.1.2

Factor $2$ out of $4 \ln (y)$ .

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

Step 8.1.3

Factor $2$ out of $2 (2 \ln (x)) + 2 (2 \ln (y))$ .

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

Step 8.1.4

Cancel the common factors.

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Step 8.1.4.1

Factor $2$ out of $2$ .

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

Step 8.1.4.2

Cancel the common factor.

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

Step 8.1.4.3

Rewrite the expression.

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

Step 8.1.4.4

Divide $2 \ln (x) + 2 \ln (y)$ by $1$ .

$\ln (x) - (2 \ln (x) + 2 \ln (y))$

Step 8.2

Apply the distributive property.

$\ln (x) - (2 \ln (x)) - (2 \ln (y))$

Step 8.3

Multiply $2$ by $- 1$ .

$\ln (x) - 2 \ln (x) - (2 \ln (y))$

Step 8.4

Multiply $2$ by $- 1$ .

$\ln (x) - 2 \ln (x) - 2 \ln (y)$

Step 9

Subtract $2 \ln (x)$ from $\ln (x)$ .

$- \ln (x) - 2 \ln (y)$