Precalculus Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

$\log_{b} (x^{3} y^{4}) - \frac{\log_{b} (z)}{2}$

Step 5

Rewrite $\log_{b} (x^{3} y^{4})$ as $\log_{b} (x^{3}) + \log_{b} (y^{4})$ .

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

Step 6

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

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

Step 7

Expand $\log_{b} (y^{4})$ by moving $4$ outside the logarithm.

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