Precalculus Examples

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

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

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

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

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

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

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

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

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

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

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

Simplify each term.

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

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

Apply the distributive property.

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

Multiply $2$ by $- 1$ .

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