Precalculus Examples

$\log_{3} (4 \sqrt{\frac{11}{s^{2} r}})$

Step 1

Rewrite $\log_{3} (4 \sqrt{\frac{11}{s^{2} r}})$ as $\log_{3} (4) + \log_{3} (\sqrt{\frac{11}{s^{2} r}})$ .

$\log_{3} (4) + \log_{3} (\sqrt{\frac{11}{s^{2} r}})$

Step 2

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{11}{s^{2} r}}$ as ${(\frac{11}{s^{2} r})}^{\frac{1}{2}}$ .

$\log_{3} (4) + \log_{3} ({(\frac{11}{s^{2} r})}^{\frac{1}{2}})$

Step 3

Expand $\log_{3} ({(\frac{11}{s^{2} r})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\log_{3} (4) + \frac{1}{2} \log_{3} (\frac{11}{s^{2} r})$

Step 4

Rewrite $\log_{3} (\frac{11}{s^{2} r})$ as $\log_{3} (11) - \log_{3} (s^{2} r)$ .

$\log_{3} (4) + \frac{1}{2} (\log_{3} (11) - \log_{3} (s^{2} r))$

Step 5

Rewrite $\log_{3} (s^{2} r)$ as $\log_{3} (s^{2}) + \log_{3} (r)$ .

$\log_{3} (4) + \frac{1}{2} (\log_{3} (11) - (\log_{3} (s^{2}) + \log_{3} (r)))$

Step 6

Expand $\log_{3} (s^{2})$ by moving $2$ outside the logarithm.

$\log_{3} (4) + \frac{1}{2} (\log_{3} (11) - (2 \log_{3} (s) + \log_{3} (r)))$

Step 7

Simplify each term.

Tap for more steps...

Step 7.1

Rewrite $\log_{3} (4)$ as $\log_{3} (2^{2})$ .

$\log_{3} (2^{2}) + \frac{1}{2} (\log_{3} (11) - (2 \log_{3} (s) + \log_{3} (r)))$

Step 7.2

Expand $\log_{3} (2^{2})$ by moving $2$ outside the logarithm.

$2 \log_{3} (2) + \frac{1}{2} (\log_{3} (11) - (2 \log_{3} (s) + \log_{3} (r)))$

Step 7.3

Simplify each term.

Tap for more steps...

Step 7.3.1

Apply the distributive property.

$2 \log_{3} (2) + \frac{1}{2} (\log_{3} (11) - (2 \log_{3} (s)) - \log_{3} (r))$

Step 7.3.2

Multiply $2$ by $- 1$ .

$2 \log_{3} (2) + \frac{1}{2} (\log_{3} (11) - 2 \log_{3} (s) - \log_{3} (r))$

Step 7.4

Apply the distributive property.

$2 \log_{3} (2) + \frac{1}{2} \log_{3} (11) + \frac{1}{2} (- 2 \log_{3} (s)) + \frac{1}{2} (- \log_{3} (r))$

Step 7.5

Simplify.

Tap for more steps...

Step 7.5.1

Combine $\frac{1}{2}$ and $\log_{3} (11)$ .

$2 \log_{3} (2) + \frac{\log_{3} (11)}{2} + \frac{1}{2} (- 2 \log_{3} (s)) + \frac{1}{2} (- \log_{3} (r))$

Step 7.5.2

Cancel the common factor of $2$ .

Tap for more steps...

Step 7.5.2.1

Factor $2$ out of $- 2 \log_{3} (s)$ .

$2 \log_{3} (2) + \frac{\log_{3} (11)}{2} + \frac{1}{2} (2 (- \log_{3} (s))) + \frac{1}{2} (- \log_{3} (r))$

Step 7.5.2.2

Cancel the common factor.

$2 \log_{3} (2) + \frac{\log_{3} (11)}{2} + \frac{1}{2} (2 (- \log_{3} (s))) + \frac{1}{2} (- \log_{3} (r))$

Step 7.5.2.3

Rewrite the expression.

$2 \log_{3} (2) + \frac{\log_{3} (11)}{2} - \log_{3} (s) + \frac{1}{2} (- \log_{3} (r))$

Step 7.5.3

Combine $\frac{1}{2}$ and $\log_{3} (r)$ .

$2 \log_{3} (2) + \frac{\log_{3} (11)}{2} - \log_{3} (s) - \frac{\log_{3} (r)}{2}$