Calculus Examples

$\log (\sqrt{a^{7} b})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{a^{7} b}$ as ${(a^{7} b)}^{\frac{1}{2}}$ .

$\log ({(a^{7} b)}^{\frac{1}{2}})$

Expand $\log ({(a^{7} b)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (a^{7} b)$

Rewrite $\log (a^{7} b)$ as $\log (a^{7}) + \log (b)$ .

$\frac{1}{2} (\log (a^{7}) + \log (b))$

Expand $\log (a^{7})$ by moving $7$ outside the logarithm.

$\frac{1}{2} (7 \log (a) + \log (b))$

Apply the distributive property.

$\frac{1}{2} (7 \log (a)) + \frac{1}{2} \log (b)$

Multiply $\frac{1}{2} (7 \log (a))$ .

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Combine $7$ and $\frac{1}{2}$ .

$\frac{7}{2} \log (a) + \frac{1}{2} \log (b)$

Combine $\frac{7}{2}$ and $\log (a)$ .

$\frac{7 \log (a)}{2} + \frac{1}{2} \log (b)$

Combine $\frac{1}{2}$ and $\log (b)$ .

$\frac{7 \log (a)}{2} + \frac{\log (b)}{2}$