Trigonometry Examples

$\log (\sqrt{\frac{x^{2}}{y}})$

Step 1

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

$\log ({(\frac{x^{2}}{y})}^{\frac{1}{2}})$

Step 2

Expand $\log ({(\frac{x^{2}}{y})}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log (\frac{x^{2}}{y})$

Step 3

Rewrite $\log (\frac{x^{2}}{y})$ as $\log (x^{2}) - \log (y)$ .

$\frac{1}{2} (\log (x^{2}) - \log (y))$

Step 4

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

$\frac{1}{2} (2 \log (x) - \log (y))$

Step 5

Apply the distributive property.

$\frac{1}{2} (2 \log (x)) + \frac{1}{2} (- \log (y))$

Step 6

Cancel the common factor of $2$ .

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Step 6.1

Factor $2$ out of $2 \log (x)$ .

$\frac{1}{2} (2 (\log (x))) + \frac{1}{2} (- \log (y))$

Step 6.2

Cancel the common factor.

$\frac{1}{2} (2 \log (x)) + \frac{1}{2} (- \log (y))$

Step 6.3

Rewrite the expression.

$\log (x) + \frac{1}{2} (- \log (y))$

Step 7

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

$\log (x) - \frac{\log (y)}{2}$