Trigonometry Examples

$\ln (\frac{1}{2 x} + 2)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$\frac{1}{2 x} + 2 = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $2$ from both sides of the equation.

$\frac{1}{2 x} = - 2$

Step 1.2.2

Find the LCD of the terms in the equation.

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Step 1.2.2.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$2 x, 1$

Step 1.2.2.2

The LCM of one and any expression is the expression.

$2 x$

Step 1.2.3

Multiply each term in $\frac{1}{2 x} = - 2$ by $2 x$ to eliminate the fractions.

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Step 1.2.3.1

Multiply each term in $\frac{1}{2 x} = - 2$ by $2 x$ .

$\frac{1}{2 x} (2 x) = - 2 (2 x)$

Step 1.2.3.2

Simplify the left side.

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Step 1.2.3.2.1

Rewrite using the commutative property of multiplication.

$2 \frac{1}{2 x} x = - 2 (2 x)$

Step 1.2.3.2.2

Cancel the common factor of $2$ .

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Step 1.2.3.2.2.1

Factor $2$ out of $2 x$ .

$2 \frac{1}{2 (x)} x = - 2 (2 x)$

Step 1.2.3.2.2.2

Cancel the common factor.

$2 \frac{1}{2 x} x = - 2 (2 x)$

Step 1.2.3.2.2.3

Rewrite the expression.

$\frac{1}{x} x = - 2 (2 x)$

Step 1.2.3.2.3

Cancel the common factor of $x$ .

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Step 1.2.3.2.3.1

Cancel the common factor.

$\frac{1}{x} x = - 2 (2 x)$

Step 1.2.3.2.3.2

Rewrite the expression.

$1 = - 2 (2 x)$

Step 1.2.3.3

Simplify the right side.

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Step 1.2.3.3.1

Multiply $2$ by $- 2$ .

$1 = - 4 x$

Step 1.2.4

Solve the equation.

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Step 1.2.4.1

Rewrite the equation as $- 4 x = 1$ .

$- 4 x = 1$

Step 1.2.4.2

Divide each term in $- 4 x = 1$ by $- 4$ and simplify.

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Step 1.2.4.2.1

Divide each term in $- 4 x = 1$ by $- 4$ .

$\frac{- 4 x}{- 4} = \frac{1}{- 4}$

Step 1.2.4.2.2

Simplify the left side.

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Step 1.2.4.2.2.1

Cancel the common factor of $- 4$ .

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Step 1.2.4.2.2.1.1

Cancel the common factor.

$\frac{- 4 x}{- 4} = \frac{1}{- 4}$

Step 1.2.4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{- 4}$

Step 1.2.4.2.3

Simplify the right side.

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Step 1.2.4.2.3.1

Move the negative in front of the fraction.

$x = - \frac{1}{4}$

Step 1.3

The vertical asymptote occurs at $x = - \frac{1}{4}$ .

Vertical Asymptote: $x = - \frac{1}{4}$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = \ln (\frac{1}{2 (1)} + 2)$

Step 2.2

Simplify the result.

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Step 2.2.1

Cancel the common factor of $1$ .

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Step 2.2.1.1

Cancel the common factor.

$f (1) = \ln (\frac{1}{2 \cdot 1} + 2)$

Step 2.2.1.2

Rewrite the expression.

$f (1) = \ln (\frac{1}{2} + 2)$

Step 2.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$f (1) = \ln (\frac{1}{2} + 2 \cdot \frac{2}{2})$

Step 2.2.3

Combine $2$ and $\frac{2}{2}$ .

$f (1) = \ln (\frac{1}{2} + \frac{2 \cdot 2}{2})$

Step 2.2.4

Combine the numerators over the common denominator.

$f (1) = \ln (\frac{1 + 2 \cdot 2}{2})$

Step 2.2.5

Simplify the numerator.

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Step 2.2.5.1

Multiply $2$ by $2$ .

$f (1) = \ln (\frac{1 + 4}{2})$

Step 2.2.5.2

Add $1$ and $4$ .

$f (1) = \ln (\frac{5}{2})$

Step 2.2.6

The final answer is $\ln (\frac{5}{2})$ .

$\ln (\frac{5}{2})$

Step 2.3

Convert $\ln (\frac{5}{2})$ to decimal.

$y = 0.91629073$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = \ln (\frac{1}{2 (2)} + 2)$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $2$ by $2$ .

$f (2) = \ln (\frac{1}{4} + 2)$

Step 3.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$f (2) = \ln (\frac{1}{4} + 2 \cdot \frac{4}{4})$

Step 3.2.3

Combine $2$ and $\frac{4}{4}$ .

$f (2) = \ln (\frac{1}{4} + \frac{2 \cdot 4}{4})$

Step 3.2.4

Combine the numerators over the common denominator.

$f (2) = \ln (\frac{1 + 2 \cdot 4}{4})$

Step 3.2.5

Simplify the numerator.

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Step 3.2.5.1

Multiply $2$ by $4$ .

$f (2) = \ln (\frac{1 + 8}{4})$

Step 3.2.5.2

Add $1$ and $8$ .

$f (2) = \ln (\frac{9}{4})$

Step 3.2.6

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

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

Step 3.3

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

$y = 0.81093021$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = \ln (\frac{1}{2 (3)} + 2)$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $2$ by $3$ .

$f (3) = \ln (\frac{1}{6} + 2)$

Step 4.2.2

To write $2$ as a fraction with a common denominator, multiply by $\frac{6}{6}$ .

$f (3) = \ln (\frac{1}{6} + 2 \cdot \frac{6}{6})$

Step 4.2.3

Combine $2$ and $\frac{6}{6}$ .

$f (3) = \ln (\frac{1}{6} + \frac{2 \cdot 6}{6})$

Step 4.2.4

Combine the numerators over the common denominator.

$f (3) = \ln (\frac{1 + 2 \cdot 6}{6})$

Step 4.2.5

Simplify the numerator.

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Step 4.2.5.1

Multiply $2$ by $6$ .

$f (3) = \ln (\frac{1 + 12}{6})$

Step 4.2.5.2

Add $1$ and $12$ .

$f (3) = \ln (\frac{13}{6})$

Step 4.2.6

The final answer is $\ln (\frac{13}{6})$ .

$\ln (\frac{13}{6})$

Step 4.3

Convert $\ln (\frac{13}{6})$ to decimal.

$y = 0.77318988$

Step 5

The log function can be graphed using the vertical asymptote at $x = - \frac{1}{4}$ and the points $(1, 0.91629073), (2, 0.81093021), (3, 0.77318988)$ .

Vertical Asymptote: $x = - \frac{1}{4}$

$\begin{array}{c} x & y \\ 1 & 0.916 \\ 2 & 0.811 \\ 3 & 0.773 \end{array}$

Step 6