Precalculus Examples

$\log (\frac{\sqrt{z^{5}}}{x^{3} y})$

Rewrite $\log (\frac{\sqrt{z^{5}}}{x^{3} y})$ as $\log (\sqrt{z^{5}}) - \log (x^{3} y)$ .

$\log (\sqrt{z^{5}}) - \log (x^{3} y)$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{z^{5}}$ as $z^{\frac{5}{2}}$ .

$\log (z^{\frac{5}{2}}) - \log (x^{3} y)$

Expand $\log (z^{\frac{5}{2}})$ by moving $\frac{5}{2}$ outside the logarithm.

$\frac{5}{2} \log (z) - \log (x^{3} y)$

Rewrite $\log (x^{3} y)$ as $\log (x^{3}) + \log (y)$ .

$\frac{5}{2} \log (z) - (\log (x^{3}) + \log (y))$

Expand $\log (x^{3})$ by moving $3$ outside the logarithm.

$\frac{5}{2} \log (z) - (3 \log (x) + \log (y))$

Simplify each term.

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Combine $\frac{5}{2}$ and $\log (z)$ .

$\frac{5 \log (z)}{2} - (3 \log (x) + \log (y))$

Apply the distributive property.

$\frac{5 \log (z)}{2} - (3 \log (x)) - \log (y)$

Multiply $3$ by $- 1$ .

$\frac{5 \log (z)}{2} - 3 \log (x) - \log (y)$