Precalculus Examples

$\log (\frac{\sqrt{y}}{x^{4} \sqrt[3]{z}})$

Step 1

Rewrite $\log (\frac{\sqrt{y}}{x^{4} \sqrt[3]{z}})$ as $\log (\sqrt{y}) - \log (x^{4} \sqrt[3]{z})$ .

$\log (\sqrt{y}) - \log (x^{4} \sqrt[3]{z})$

Step 2

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

$\log (y^{\frac{1}{2}}) - \log (x^{4} \sqrt[3]{z})$

Step 3

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{z}$ as $z^{\frac{1}{3}}$ .

$\log (y^{\frac{1}{2}}) - \log (x^{4} z^{\frac{1}{3}})$

Step 4

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

$\frac{1}{2} \log (y) - \log (x^{4} z^{\frac{1}{3}})$

Step 5

Rewrite $\log (x^{4} z^{\frac{1}{3}})$ as $\log (x^{4}) + \log (z^{\frac{1}{3}})$ .

$\frac{1}{2} \log (y) - (\log (x^{4}) + \log (z^{\frac{1}{3}}))$

Step 6

Expand $\log (x^{4})$ by moving $4$ outside the logarithm.

$\frac{1}{2} \log (y) - (4 \log (x) + \log (z^{\frac{1}{3}}))$

Step 7

Expand $\log (z^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{2} \log (y) - (4 \log (x) + \frac{1}{3} \log (z))$

Step 8

Simplify each term.

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Step 8.1

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

$\frac{\log (y)}{2} - (4 \log (x) + \frac{1}{3} \log (z))$

Step 8.2

Combine $\frac{1}{3}$ and $\log (z)$ .

$\frac{\log (y)}{2} - (4 \log (x) + \frac{\log (z)}{3})$

Step 8.3

Apply the distributive property.

$\frac{\log (y)}{2} - (4 \log (x)) - \frac{\log (z)}{3}$

Step 8.4

Multiply $4$ by $- 1$ .

$\frac{\log (y)}{2} - 4 \log (x) - \frac{\log (z)}{3}$