Precalculus Examples

$\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}}}{\sqrt{x^{3} - x}})$

Step 1

Rewrite $\log_{5} (\frac{x^{2} {(1 - 5 x)}^{\frac{3}{2}}}{\sqrt{x^{3} - x}})$ as $\log_{5} (x^{2} {(1 - 5 x)}^{\frac{3}{2}}) - \log_{5} (\sqrt{x^{3} - x})$ .

$\log_{5} (x^{2} {(1 - 5 x)}^{\frac{3}{2}}) - \log_{5} (\sqrt{x^{3} - x})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{3} - x}$ as ${(x^{3} - x)}^{\frac{1}{2}}$ .

$\log_{5} (x^{2} {(1 - 5 x)}^{\frac{3}{2}}) - \log_{5} ({(x^{3} - x)}^{\frac{1}{2}})$

Step 3

Expand $\log_{5} ({(x^{3} - x)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{5} (x^{2} {(1 - 5 x)}^{\frac{3}{2}}) - (\frac{1}{2} \log_{5} (x^{3} - x))$

Step 4

Combine $\frac{1}{2}$ and $\log_{5} (x^{3} - x)$ .

$\log_{5} (x^{2} {(1 - 5 x)}^{\frac{3}{2}}) - \frac{\log_{5} (x^{3} - x)}{2}$

Step 5

Rewrite $\log_{5} (x^{2} {(1 - 5 x)}^{\frac{3}{2}})$ as $\log_{5} (x^{2}) + \log_{5} ({(1 - 5 x)}^{\frac{3}{2}})$ .

$\log_{5} (x^{2}) + \log_{5} ({(1 - 5 x)}^{\frac{3}{2}}) - \frac{\log_{5} (x^{3} - x)}{2}$

Step 6

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

$2 \log_{5} (x) + \log_{5} ({(1 - 5 x)}^{\frac{3}{2}}) - \frac{\log_{5} (x^{3} - x)}{2}$

Step 7

Expand $\log_{5} ({(1 - 5 x)}^{\frac{3}{2}})$ by moving $\frac{3}{2}$ outside the logarithm.

$2 \log_{5} (x) + \frac{3}{2} \log_{5} (1 - 5 x) - \frac{\log_{5} (x^{3} - x)}{2}$

Step 8

Combine $\frac{3}{2}$ and $\log_{5} (1 - 5 x)$ .

$2 \log_{5} (x) + \frac{3 \log_{5} (1 - 5 x)}{2} - \frac{\log_{5} (x^{3} - x)}{2}$

Step 9

To write $2 \log_{5} (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$2 \log_{5} (x) \cdot \frac{2}{2} + \frac{3 \log_{5} (1 - 5 x)}{2} - \frac{\log_{5} (x^{3} - x)}{2}$

Step 10

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

$\frac{2 \log_{5} (x) \cdot 2}{2} + \frac{3 \log_{5} (1 - 5 x)}{2} - \frac{\log_{5} (x^{3} - x)}{2}$

Step 11

Combine the numerators over the common denominator.

$\frac{2 \log_{5} (x) \cdot 2 + 3 \log_{5} (1 - 5 x)}{2} - \frac{\log_{5} (x^{3} - x)}{2}$

Step 12

Combine the numerators over the common denominator.

$\frac{2 \log_{5} (x) \cdot 2 + 3 \log_{5} (1 - 5 x) - \log_{5} (x^{3} - x)}{2}$

Step 13

Multiply $2$ by $2$ .

$\frac{4 \log_{5} (x) + 3 \log_{5} (1 - 5 x) - \log_{5} (x^{3} - x)}{2}$