Precalculus Examples

$\ln (\frac{\sqrt{x}}{8 y^{3}})$

Step 1

Rewrite $\ln (\frac{\sqrt{x}}{8 y^{3}})$ as $\ln (\sqrt{x}) - \ln (8 y^{3})$ .

$\ln (\sqrt{x}) - \ln (8 y^{3})$

Step 2

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

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

Step 3

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

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

Step 4

Rewrite $\ln (8 y^{3})$ as $\ln (8) + \ln (y^{3})$ .

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

Step 5

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

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

Step 6

Simplify each term.

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Step 6.1

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

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

Step 6.2

Simplify each term.

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Step 6.2.1

Rewrite $\ln (8)$ as $\ln (2^{3})$ .

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

Step 6.2.2

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

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

Step 6.3

Apply the distributive property.

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

Step 6.4

Multiply $3$ by $- 1$ .

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

Step 6.5

Multiply $3$ by $- 1$ .

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