Algebra Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

Apply the distributive property.

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

Step 7

Apply the distributive property.

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

Step 8

Simplify.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

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

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