Calculus Examples

$\ln (\frac{\sqrt{a b^{8}}}{c})$

Rewrite $\ln (\frac{\sqrt{a b^{8}}}{c})$ as $\ln (\sqrt{a b^{8}}) - \ln (c)$ .

$\ln (\sqrt{a b^{8}}) - \ln (c)$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a b^{8}}$ as ${(a b^{8})}^{\frac{1}{2}}$ .

$\ln ({(a b^{8})}^{\frac{1}{2}}) - \ln (c)$

Expand $\ln ({(a b^{8})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \ln (a b^{8}) - \ln (c)$

Rewrite $\ln (a b^{8})$ as $\ln (a) + \ln (b^{8})$ .

$\frac{1}{2} (\ln (a) + \ln (b^{8})) - \ln (c)$

Expand $\ln (b^{8})$ by moving $8$ outside the logarithm.

$\frac{1}{2} (\ln (a) + 8 \ln (b)) - \ln (c)$

Simplify each term.

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Apply the distributive property.

$\frac{1}{2} \ln (a) + \frac{1}{2} (8 \ln (b)) - \ln (c)$

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

$\frac{\ln (a)}{2} + \frac{1}{2} (8 \ln (b)) - \ln (c)$

Cancel the common factor of $2$ .

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Factor $2$ out of $8 \ln (b)$ .

$\frac{\ln (a)}{2} + \frac{1}{2} (2 (4 \ln (b))) - \ln (c)$

Cancel the common factor.

$\frac{\ln (a)}{2} + \frac{1}{2} (2 (4 \ln (b))) - \ln (c)$

Rewrite the expression.

$\frac{\ln (a)}{2} + 4 \ln (b) - \ln (c)$