Precalculus Examples

$\log (\sqrt{\frac{2 (r s t)}{5 w}})$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\frac{2 (r s t)}{5 w}}$ as ${(\frac{2 (r s t)}{5 w})}^{\frac{1}{2}}$ .

$\log ({(\frac{2 (r s t)}{5 w})}^{\frac{1}{2}})$

Step 2

Expand $\log ({(\frac{2 (r s t)}{5 w})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (\frac{2 (r s t)}{5 w})$

Step 3

Remove parentheses.

$\frac{1}{2} \log (\frac{2 (r s) t}{5 w})$

Step 4

Remove parentheses.

$\frac{1}{2} \log (\frac{2 r s t}{5 w})$

Step 5

Rewrite $\log (\frac{2 r s t}{5 w})$ as $\log (2 r s t) - \log (5 w)$ .

$\frac{1}{2} (\log (2 r s t) - \log (5 w))$

Step 6

Rewrite $\log (2 r s t)$ as $\log (2 r s) + \log (t)$ .

$\frac{1}{2} (\log (2 r s) + \log (t) - \log (5 w))$

Step 7

Rewrite $\log (2 r s)$ as $\log (2 r) + \log (s)$ .

$\frac{1}{2} (\log (2 r) + \log (s) + \log (t) - \log (5 w))$

Step 8

Rewrite $\log (2 r)$ as $\log (2) + \log (r)$ .

$\frac{1}{2} (\log (2) + \log (r) + \log (s) + \log (t) - \log (5 w))$

Step 9

Rewrite $\log (5 w)$ as $\log (5) + \log (w)$ .

$\frac{1}{2} (\log (2) + \log (r) + \log (s) + \log (t) - (\log (5) + \log (w)))$

Step 10

Apply the distributive property.

$\frac{1}{2} (\log (2) + \log (r) + \log (s) + \log (t) - \log (5) - \log (w))$

Step 11

Apply the distributive property.

$\frac{1}{2} \log (2) + \frac{1}{2} \log (r) + \frac{1}{2} \log (s) + \frac{1}{2} \log (t) + \frac{1}{2} (- \log (5)) + \frac{1}{2} (- \log (w))$

Step 12

Simplify.

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Step 12.1

Combine $\frac{1}{2}$ and $\log (2)$ .

$\frac{\log (2)}{2} + \frac{1}{2} \log (r) + \frac{1}{2} \log (s) + \frac{1}{2} \log (t) + \frac{1}{2} (- \log (5)) + \frac{1}{2} (- \log (w))$

Step 12.2

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

$\frac{\log (2)}{2} + \frac{\log (r)}{2} + \frac{1}{2} \log (s) + \frac{1}{2} \log (t) + \frac{1}{2} (- \log (5)) + \frac{1}{2} (- \log (w))$

Step 12.3

Combine $\frac{1}{2}$ and $\log (s)$ .

$\frac{\log (2)}{2} + \frac{\log (r)}{2} + \frac{\log (s)}{2} + \frac{1}{2} \log (t) + \frac{1}{2} (- \log (5)) + \frac{1}{2} (- \log (w))$

Step 12.4

Combine $\frac{1}{2}$ and $\log (t)$ .

$\frac{\log (2)}{2} + \frac{\log (r)}{2} + \frac{\log (s)}{2} + \frac{\log (t)}{2} + \frac{1}{2} (- \log (5)) + \frac{1}{2} (- \log (w))$

Step 12.5

Combine $\frac{1}{2}$ and $\log (5)$ .

$\frac{\log (2)}{2} + \frac{\log (r)}{2} + \frac{\log (s)}{2} + \frac{\log (t)}{2} - \frac{\log (5)}{2} + \frac{1}{2} (- \log (w))$

Step 12.6

Combine $\frac{1}{2}$ and $\log (w)$ .

$\frac{\log (2)}{2} + \frac{\log (r)}{2} + \frac{\log (s)}{2} + \frac{\log (t)}{2} - \frac{\log (5)}{2} - \frac{\log (w)}{2}$