Precalculus Examples

$\log_{b} (\sqrt[3]{\frac{x + 3}{x}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{\frac{x + 3}{x}}$ as ${(\frac{x + 3}{x})}^{\frac{1}{3}}$ .

$\log_{b} ({(\frac{x + 3}{x})}^{\frac{1}{3}})$

Step 2

Expand $\log_{b} ({(\frac{x + 3}{x})}^{\frac{1}{3}})$ by moving $\frac{1}{3}$ outside the logarithm.

$\frac{1}{3} \log_{b} (\frac{x + 3}{x})$

Step 3

Rewrite $\log_{b} (\frac{x + 3}{x})$ as $\log_{b} (x + 3) - \log_{b} (x)$ .

$\frac{1}{3} (\log_{b} (x + 3) - \log_{b} (x))$

Step 4

Apply the distributive property.

$\frac{1}{3} \log_{b} (x + 3) + \frac{1}{3} (- \log_{b} (x))$

Step 5

Combine $\frac{1}{3}$ and $\log_{b} (x + 3)$ .

$\frac{\log_{b} (x + 3)}{3} + \frac{1}{3} (- \log_{b} (x))$

Step 6

Combine $\frac{1}{3}$ and $\log_{b} (x)$ .

$\frac{\log_{b} (x + 3)}{3} - \frac{\log_{b} (x)}{3}$

Step 7

Combine the numerators over the common denominator.

$\frac{\log_{b} (x + 3) - \log_{b} (x)}{3}$