Precalculus Examples

$\log_{b} (\frac{\sqrt{x^{5}} (y^{2})}{z^{9}})$

Step 1

Rewrite $\log_{b} (\frac{\sqrt{x^{5}} (y^{2})}{z^{9}})$ as $\log_{b} (\sqrt{x^{5}} (y^{2})) - \log_{b} (z^{9})$ .

$\log_{b} (\sqrt{x^{5}} (y^{2})) - \log_{b} (z^{9})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{5}}$ as $x^{\frac{5}{2}}$ .

$\log_{b} (x^{\frac{5}{2}} y^{2}) - \log_{b} (z^{9})$

Step 3

Expand $\log_{b} (z^{9})$ by moving $9$ outside the logarithm.

$\log_{b} (x^{\frac{5}{2}} y^{2}) - (9 \log_{b} (z))$

Step 4

Multiply $9$ by $- 1$ .

$\log_{b} (x^{\frac{5}{2}} y^{2}) - 9 \log_{b} (z)$

Step 5

Rewrite $\log_{b} (x^{\frac{5}{2}} y^{2})$ as $\log_{b} (x^{\frac{5}{2}}) + \log_{b} (y^{2})$ .

$\log_{b} (x^{\frac{5}{2}}) + \log_{b} (y^{2}) - 9 \log_{b} (z)$

Step 6

Expand $\log_{b} (x^{\frac{5}{2}})$ by moving $\frac{5}{2}$ outside the logarithm.

$\frac{5}{2} \log_{b} (x) + \log_{b} (y^{2}) - 9 \log_{b} (z)$

Step 7

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

$\frac{5}{2} \log_{b} (x) + 2 \log_{b} (y) - 9 \log_{b} (z)$

Step 8

Combine $\frac{5}{2}$ and $\log_{b} (x)$ .

$\frac{5 \log_{b} (x)}{2} + 2 \log_{b} (y) - 9 \log_{b} (z)$