Precalculus Examples

$\log (\sqrt{\frac{z y^{7}}{x^{5}}})$

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{z y^{7}}{x^{5}}}$ as ${(\frac{z y^{7}}{x^{5}})}^{\frac{1}{2}}$ .

$\log ({(\frac{z y^{7}}{x^{5}})}^{\frac{1}{2}})$

Expand $\log ({(\frac{z y^{7}}{x^{5}})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (\frac{z y^{7}}{x^{5}})$

Rewrite $\log (\frac{z y^{7}}{x^{5}})$ as $\log (z y^{7}) - \log (x^{5})$ .

$\frac{1}{2} (\log (z y^{7}) - \log (x^{5}))$

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

$\frac{1}{2} (\log (z y^{7}) - (5 \log (x)))$

Multiply $5$ by $- 1$ .

$\frac{1}{2} (\log (z y^{7}) - 5 \log (x))$

Rewrite $\log (z y^{7})$ as $\log (z) + \log (y^{7})$ .

$\frac{1}{2} (\log (z) + \log (y^{7}) - 5 \log (x))$

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

$\frac{1}{2} (\log (z) + 7 \log (y) - 5 \log (x))$

Apply the distributive property.

$\frac{1}{2} \log (z) + \frac{1}{2} (7 \log (y)) + \frac{1}{2} (- 5 \log (x))$

Simplify.

Tap for more steps...

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

$\frac{\log (z)}{2} + \frac{1}{2} (7 \log (y)) + \frac{1}{2} (- 5 \log (x))$

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

Tap for more steps...

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

$\frac{\log (z)}{2} + \frac{7}{2} \log (y) + \frac{1}{2} (- 5 \log (x))$

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

$\frac{\log (z)}{2} + \frac{7 \log (y)}{2} + \frac{1}{2} (- 5 \log (x))$

Multiply $\frac{1}{2} (- 5 \log (x))$ .

Tap for more steps...

Combine $- 5$ and $\frac{1}{2}$ .

$\frac{\log (z)}{2} + \frac{7 \log (y)}{2} + \frac{- 5}{2} \log (x)$

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

$\frac{\log (z)}{2} + \frac{7 \log (y)}{2} + \frac{- 5 \log (x)}{2}$

Move the negative in front of the fraction.

$\frac{\log (z)}{2} + \frac{7 \log (y)}{2} - \frac{5 \log (x)}{2}$