Trigonometry Examples

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

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

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

Step 1.2

Solve for $x$ .

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Step 1.2.1

Set the numerator equal to zero.

$x^{2} = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

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

$x = \pm \sqrt{0}$

Step 1.2.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.2.2.1

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

$x = \pm \sqrt{0^{2}}$

Step 1.2.2.2.2

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm 0$

Step 1.2.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

Step 1.3

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 2$ .

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Step 2.1

Replace the variable $x$ with $2$ in the expression.

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

Step 2.2

Simplify the result.

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Step 2.2.1

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

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Step 2.2.1.1

Factor $3$ out of $3 {(2)}^{2}$ .

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

Step 2.2.1.2

Cancel the common factors.

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Step 2.2.1.2.1

Factor $3$ out of $12$ .

$f (2) = \log (\frac{3 {(2)}^{2}}{3 \cdot 4})$

Step 2.2.1.2.2

Cancel the common factor.

$f (2) = \log (\frac{3 {(2)}^{2}}{3 \cdot 4})$

Step 2.2.1.2.3

Rewrite the expression.

$f (2) = \log (\frac{{(2)}^{2}}{4})$

Step 2.2.2

Simplify the expression.

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Step 2.2.2.1

Raise $2$ to the power of $2$ .

$f (2) = \log (\frac{4}{4})$

Step 2.2.2.2

Divide $4$ by $4$ .

$f (2) = \log (1)$

Step 2.2.3

Logarithm base $10$ of $1$ is $0$ .

$f (2) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 1$ .

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Step 3.1

Replace the variable $x$ with $1$ in the expression.

$f (1) = \log (\frac{3 {(1)}^{2}}{12})$

Step 3.2

Simplify the result.

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Step 3.2.1

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

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Step 3.2.1.1

Factor $3$ out of $3 {(1)}^{2}$ .

$f (1) = \log (\frac{3 ({(1)}^{2})}{12})$

Step 3.2.1.2

Cancel the common factors.

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Step 3.2.1.2.1

Factor $3$ out of $12$ .

$f (1) = \log (\frac{3 {(1)}^{2}}{3 \cdot 4})$

Step 3.2.1.2.2

Cancel the common factor.

$f (1) = \log (\frac{3 {(1)}^{2}}{3 \cdot 4})$

Step 3.2.1.2.3

Rewrite the expression.

$f (1) = \log (\frac{{(1)}^{2}}{4})$

Step 3.2.2

One to any power is one.

$f (1) = \log (\frac{1}{4})$

Step 3.2.3

The final answer is $\log (\frac{1}{4})$ .

$\log (\frac{1}{4})$

Step 3.3

Convert $\log (\frac{1}{4})$ to decimal.

$y = - 0.60205999$

Step 4

Find the point at $x = 3$ .

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Step 4.1

Replace the variable $x$ with $3$ in the expression.

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

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $3$ by ${(3)}^{2}$ by adding the exponents.

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Step 4.2.1.1

Multiply $3$ by ${(3)}^{2}$ .

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Step 4.2.1.1.1

Raise $3$ to the power of $1$ .

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

Step 4.2.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$f (3) = \log (\frac{3^{1 + 2}}{12})$

Step 4.2.1.2

Add $1$ and $2$ .

$f (3) = \log (\frac{3^{3}}{12})$

Step 4.2.2

Raise $3$ to the power of $3$ .

$f (3) = \log (\frac{27}{12})$

Step 4.2.3

Cancel the common factor of $27$ and $12$ .

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Step 4.2.3.1

Factor $3$ out of $27$ .

$f (3) = \log (\frac{3 (9)}{12})$

Step 4.2.3.2

Cancel the common factors.

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Step 4.2.3.2.1

Factor $3$ out of $12$ .

$f (3) = \log (\frac{3 \cdot 9}{3 \cdot 4})$

Step 4.2.3.2.2

Cancel the common factor.

$f (3) = \log (\frac{3 \cdot 9}{3 \cdot 4})$

Step 4.2.3.2.3

Rewrite the expression.

$f (3) = \log (\frac{9}{4})$

Step 4.2.4

The final answer is $\log (\frac{9}{4})$ .

$\log (\frac{9}{4})$

Step 4.3

Convert $\log (\frac{9}{4})$ to decimal.

$y = 0.35218251$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(2, 0), (1, - 0.60205999), (3, 0.35218251)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & - 0.602 \\ 2 & 0 \\ 3 & 0.352 \end{array}$

Step 6