Precalculus Examples

$\log_{a} (\frac{x^{3} (y^{2})}{\sqrt{w}})$

Step 1

Rewrite $\log_{a} (\frac{x^{3} (y^{2})}{\sqrt{w}})$ as $\log_{a} (x^{3} (y^{2})) - \log_{a} (\sqrt{w})$ .

$\log_{a} (x^{3} (y^{2})) - \log_{a} (\sqrt{w})$

Step 2

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

$\log_{a} (x^{3} (y^{2})) - \log_{a} (w^{\frac{1}{2}})$

Step 3

Expand $\log_{a} (w^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{a} (x^{3} (y^{2})) - (\frac{1}{2} \log_{a} (w))$

Step 4

Combine $\frac{1}{2}$ and $\log_{a} (w)$ .

$\log_{a} (x^{3} (y^{2})) - \frac{\log_{a} (w)}{2}$

Step 5

Multiply $x^{3}$ by $y^{2}$ .

$\log_{a} (x^{3} y^{2}) - \frac{\log_{a} (w)}{2}$

Step 6

Rewrite $\log_{a} (x^{3} y^{2})$ as $\log_{a} (x^{3}) + \log_{a} (y^{2})$ .

$\log_{a} (x^{3}) + \log_{a} (y^{2}) - \frac{\log_{a} (w)}{2}$

Step 7

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

$3 \log_{a} (x) + \log_{a} (y^{2}) - \frac{\log_{a} (w)}{2}$

Step 8

Expand $\log_{a} (y^{2})$ by moving $2$ outside the logarithm.

$3 \log_{a} (x) + 2 \log_{a} (y) - \frac{\log_{a} (w)}{2}$