Precalculus Examples

$\log_{6} (\frac{125 \sqrt[5]{r - 4}}{\sqrt{r}})$

Step 1

Rewrite $\log_{6} (\frac{125 \sqrt[5]{r - 4}}{\sqrt{r}})$ as $\log_{6} (125 \sqrt[5]{r - 4}) - \log_{6} (\sqrt{r})$ .

$\log_{6} (125 \sqrt[5]{r - 4}) - \log_{6} (\sqrt{r})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[5]{r - 4}$ as ${(r - 4)}^{\frac{1}{5}}$ .

$\log_{6} (125 {(r - 4)}^{\frac{1}{5}}) - \log_{6} (\sqrt{r})$

Step 3

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

$\log_{6} (125 {(r - 4)}^{\frac{1}{5}}) - \log_{6} (r^{\frac{1}{2}})$

Step 4

Expand $\log_{6} (r^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{6} (125 {(r - 4)}^{\frac{1}{5}}) - (\frac{1}{2} \log_{6} (r))$

Step 5

Combine $\frac{1}{2}$ and $\log_{6} (r)$ .

$\log_{6} (125 {(r - 4)}^{\frac{1}{5}}) - \frac{\log_{6} (r)}{2}$

Step 6

Rewrite $\log_{6} (125 {(r - 4)}^{\frac{1}{5}})$ as $\log_{6} (125) + \log_{6} ({(r - 4)}^{\frac{1}{5}})$ .

$\log_{6} (125) + \log_{6} ({(r - 4)}^{\frac{1}{5}}) - \frac{\log_{6} (r)}{2}$

Step 7

Expand $\log_{6} ({(r - 4)}^{\frac{1}{5}})$ by moving $\frac{1}{5}$ outside the logarithm.

$\log_{6} (125) + \frac{1}{5} \log_{6} (r - 4) - \frac{\log_{6} (r)}{2}$

Step 8

Simplify each term.

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Step 8.1

Rewrite $\log_{6} (125)$ as $\log_{6} (5^{3})$ .

$\log_{6} (5^{3}) + \frac{1}{5} \log_{6} (r - 4) - \frac{\log_{6} (r)}{2}$

Step 8.2

Expand $\log_{6} (5^{3})$ by moving $3$ outside the logarithm.

$3 \log_{6} (5) + \frac{1}{5} \log_{6} (r - 4) - \frac{\log_{6} (r)}{2}$

Step 8.3

Combine $\frac{1}{5}$ and $\log_{6} (r - 4)$ .

$3 \log_{6} (5) + \frac{\log_{6} (r - 4)}{5} - \frac{\log_{6} (r)}{2}$

Step 9

To write $3 \log_{6} (5)$ as a fraction with a common denominator, multiply by $\frac{5}{5}$ .

$3 \log_{6} (5) \cdot \frac{5}{5} + \frac{\log_{6} (r - 4)}{5} - \frac{\log_{6} (r)}{2}$

Step 10

Combine $3 \log_{6} (5)$ and $\frac{5}{5}$ .

$\frac{3 \log_{6} (5) \cdot 5}{5} + \frac{\log_{6} (r - 4)}{5} - \frac{\log_{6} (r)}{2}$

Step 11

Combine the numerators over the common denominator.

$\frac{3 \log_{6} (5) \cdot 5 + \log_{6} (r - 4)}{5} - \frac{\log_{6} (r)}{2}$

Step 12

Multiply $5$ by $3$ .

$\frac{15 \log_{6} (5) + \log_{6} (r - 4)}{5} - \frac{\log_{6} (r)}{2}$