Precalculus Examples

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

Step 1

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

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

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{x z^{2}}{y^{\frac{1}{2}}}}$ as ${(\frac{x z^{2}}{y^{\frac{1}{2}}})}^{\frac{1}{3}}$ .

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

Step 3

Expand $\log ({(\frac{x z^{2}}{y^{\frac{1}{2}}})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

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

Step 4

Rewrite $\log (\frac{x z^{2}}{y^{\frac{1}{2}}})$ as $\log (x z^{2}) - \log (y^{\frac{1}{2}})$ .

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

Step 5

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

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

Step 6

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

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

Step 7

Rewrite $\log (x z^{2})$ as $\log (x) + \log (z^{2})$ .

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

Step 8

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

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

Step 9

Apply the distributive property.

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

Step 10

Simplify.

Tap for more steps...

Step 10.1

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

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

Step 10.2

Multiply $\frac{1}{3} (2 \log (z))$ .

Tap for more steps...

Step 10.2.1

Combine $2$ and $\frac{1}{3}$ .

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

Step 10.2.2

Combine $\frac{2}{3}$ and $\log (z)$ .

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

Step 10.3

Multiply $\frac{1}{3} (- \frac{\log (y)}{2})$ .

Tap for more steps...

Step 10.3.1

Multiply $\frac{1}{3}$ by $\frac{\log (y)}{2}$ .

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

Step 10.3.2

Multiply $3$ by $2$ .

$\frac{\log (x)}{3} + \frac{2 \log (z)}{3} - \frac{\log (y)}{6}$