Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Multiply $3$ by $- 1$ .

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

Step 5

Rewrite $\log (x^{4} z^{\frac{1}{2}})$ as $\log (x^{4}) + \log (z^{\frac{1}{2}})$ .

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

Step 6

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

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

Step 7

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

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

Step 8

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

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