Trigonometry Examples

$\log (\frac{3 x^{2}}{12})$

Step 1

Set the argument in $\log (\frac{3 x^{2}}{12})$ greater than $0$ to find where the expression is defined.

$\frac{3 x^{2}}{12} > 0$

Step 2

Solve for $x$ .

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Step 2.1

Cancel the common factor of $3$ and $12$ .

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Step 2.1.1

Factor $3$ out of $3 x^{2}$ .

$\frac{3 (x^{2})}{12} > 0$

Step 2.1.2

Cancel the common factors.

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Step 2.1.2.1

Factor $3$ out of $12$ .

$\frac{3 x^{2}}{3 \cdot 4} > 0$

Step 2.1.2.2

Cancel the common factor.

$\frac{3 x^{2}}{3 \cdot 4} > 0$

Step 2.1.2.3

Rewrite the expression.

$\frac{x^{2}}{4} > 0$

Step 2.2

Multiply both sides by $4$ .

$\frac{x^{2}}{4} \cdot 4 > 0 \cdot 4$

Step 2.3

Simplify.

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Step 2.3.1

Simplify the left side.

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Step 2.3.1.1

Cancel the common factor of $4$ .

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Step 2.3.1.1.1

Cancel the common factor.

$\frac{x^{2}}{4} \cdot 4 > 0 \cdot 4$

Step 2.3.1.1.2

Rewrite the expression.

$x^{2} > 0 \cdot 4$

Step 2.3.2

Simplify the right side.

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Step 2.3.2.1

Multiply $0$ by $4$ .

$x^{2} > 0$

Step 2.4

Solve for $x$ .

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Step 2.4.1

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{x^{2}} > \sqrt{0}$

Step 2.4.2

Simplify the equation.

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Step 2.4.2.1

Simplify the left side.

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Step 2.4.2.1.1

Pull terms out from under the radical.

$| x | > \sqrt{0}$

Step 2.4.2.2

Simplify the right side.

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Step 2.4.2.2.1

Simplify $\sqrt{0}$ .

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Step 2.4.2.2.1.1

Rewrite $0$ as $0^{2}$ .

$| x | > \sqrt{0^{2}}$

Step 2.4.2.2.1.2

Pull terms out from under the radical.

$| x | > | 0 |$

Step 2.4.2.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $0$ is $0$ .

$| x | > 0$

Step 2.4.3

Write $| x | > 0$ as a piecewise.

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Step 2.4.3.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$x \geq 0$

Step 2.4.3.2

In the piece where $x$ is non-negative, remove the absolute value.

$x > 0$

Step 2.4.3.3

To find the interval for the second piece, find where the inside of the absolute value is negative.

$x < 0$

Step 2.4.3.4

In the piece where $x$ is negative, remove the absolute value and multiply by $- 1$ .

$- x > 0$

Step 2.4.3.5

Write as a piecewise.

${\begin{cases} x > 0 & x \geq 0 \\ - x > 0 & x < 0 \end{cases}$

Step 2.4.4

Find the intersection of $x > 0$ and $x \geq 0$ .

$x > 0$

Step 2.4.5

Divide each term in $- x > 0$ by $- 1$ and simplify.

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Step 2.4.5.1

Divide each term in $- x > 0$ by $- 1$ . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.

$\frac{- x}{- 1} < \frac{0}{- 1}$

Step 2.4.5.2

Simplify the left side.

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Step 2.4.5.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} < \frac{0}{- 1}$

Step 2.4.5.2.2

Divide $x$ by $1$ .

$x < \frac{0}{- 1}$

Step 2.4.5.3

Simplify the right side.

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Step 2.4.5.3.1

Divide $0$ by $- 1$ .

$x < 0$

Step 2.4.6

Find the union of the solutions.

$x < 0$ or $x > 0$

Step 3

The domain is all values of $x$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${x | x \neq 0}$

Step 4