Trigonometry Examples

$\log_{11} (\frac{\sqrt{x + y}}{x^{2} y})$

Step 1

Rewrite $\log_{11} (\frac{\sqrt{x + y}}{x^{2} y})$ as $\log_{11} (\sqrt{x + y}) - \log_{11} (x^{2} y)$ .

$\log_{11} (\sqrt{x + y}) - \log_{11} (x^{2} y)$

Step 2

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

$\log_{11} ({(x + y)}^{\frac{1}{2}}) - \log_{11} (x^{2} y)$

Step 3

Expand $\log_{11} ({(x + y)}^{\frac{1}{2}})$ by moving $\frac{1}{2}$ outside the logarithm.

$\frac{1}{2} \log_{11} (x + y) - \log_{11} (x^{2} y)$

Step 4

Rewrite $\log_{11} (x^{2} y)$ as $\log_{11} (x^{2}) + \log_{11} (y)$ .

$\frac{1}{2} \log_{11} (x + y) - (\log_{11} (x^{2}) + \log_{11} (y))$

Step 5

Expand $\log_{11} (x^{2})$ by moving $2$ outside the logarithm.

$\frac{1}{2} \log_{11} (x + y) - (2 \log_{11} (x) + \log_{11} (y))$

Step 6

Simplify each term.

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Step 6.1

Combine $\frac{1}{2}$ and $\log_{11} (x + y)$ .

$\frac{\log_{11} (x + y)}{2} - (2 \log_{11} (x) + \log_{11} (y))$

Step 6.2

Apply the distributive property.

$\frac{\log_{11} (x + y)}{2} - (2 \log_{11} (x)) - \log_{11} (y)$

Step 6.3

Multiply $2$ by $- 1$ .

$\frac{\log_{11} (x + y)}{2} - 2 \log_{11} (x) - \log_{11} (y)$

Step 7

To write $- 2 \log_{11} (x)$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{\log_{11} (x + y)}{2} - 2 \log_{11} (x) \cdot \frac{2}{2} - \log_{11} (y)$

Step 8

Combine $- 2 \log_{11} (x)$ and $\frac{2}{2}$ .

$\frac{\log_{11} (x + y)}{2} + \frac{- 2 \log_{11} (x) \cdot 2}{2} - \log_{11} (y)$

Step 9

Combine the numerators over the common denominator.

$\frac{\log_{11} (x + y) - 2 \log_{11} (x) \cdot 2}{2} - \log_{11} (y)$

Step 10

Multiply $2$ by $- 2$ .

$\frac{\log_{11} (x + y) - 4 \log_{11} (x)}{2} - \log_{11} (y)$