Trigonometry Examples

$\log (\frac{x (x + 4)}{{(x + 2)}^{2}})$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$\frac{x (x + 4)}{{(x + 2)}^{2}} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Set the numerator equal to zero.

$x (x + 4) = 0$

Step 1.2.2

Solve the equation for $x$ .

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Step 1.2.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$x + 4 = 0$

Step 1.2.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.2.3

Set $x + 4$ equal to $0$ and solve for $x$ .

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Step 1.2.2.3.1

Set $x + 4$ equal to $0$ .

$x + 4 = 0$

Step 1.2.2.3.2

Subtract $4$ from both sides of the equation.

$x = - 4$

Step 1.2.2.4

The final solution is all the values that make $x (x + 4) = 0$ true.

$x = 0, - 4$

Step 1.3

The vertical asymptote occurs at $x = 0, x = - 4$ .

Vertical Asymptote: $x = 0, x = - 4$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

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

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $1 + 4$ by $1$ .

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

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

Add $1$ and $2$ .

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

Step 2.2.2.2

Raise $3$ to the power of $2$ .

$f (1) = \log (\frac{1 + 4}{9})$

Step 2.2.3

Add $1$ and $4$ .

$f (1) = \log (\frac{5}{9})$

Step 2.2.4

The final answer is $\log (\frac{5}{9})$ .

$\log (\frac{5}{9})$

Step 2.3

Convert $\log (\frac{5}{9})$ to decimal.

$y = - 0.2552725$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Add $2$ and $4$ .

$f (2) = \log (\frac{2 \cdot 6}{{(2 + 2)}^{2}})$

Step 3.2.2

Simplify the denominator.

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Step 3.2.2.1

Add $2$ and $2$ .

$f (2) = \log (\frac{2 \cdot 6}{4^{2}})$

Step 3.2.2.2

Raise $4$ to the power of $2$ .

$f (2) = \log (\frac{2 \cdot 6}{16})$

Step 3.2.3

Reduce the expression by cancelling the common factors.

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Step 3.2.3.1

Multiply $2$ by $6$ .

$f (2) = \log (\frac{12}{16})$

Step 3.2.3.2

Cancel the common factor of $12$ and $16$ .

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Step 3.2.3.2.1

Factor $4$ out of $12$ .

$f (2) = \log (\frac{4 (3)}{16})$

Step 3.2.3.2.2

Cancel the common factors.

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Step 3.2.3.2.2.1

Factor $4$ out of $16$ .

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

Step 3.2.3.2.2.2

Cancel the common factor.

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

Step 3.2.3.2.2.3

Rewrite the expression.

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

Step 3.2.4

The final answer is $\log (\frac{3}{4})$ .

$\log (\frac{3}{4})$

Step 3.3

Convert $\log (\frac{3}{4})$ to decimal.

$y = - 0.12493873$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

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

Step 4.2

Simplify the result.

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Step 4.2.1

Add $3$ and $4$ .

$f (3) = \log (\frac{3 \cdot 7}{{(3 + 2)}^{2}})$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

Add $3$ and $2$ .

$f (3) = \log (\frac{3 \cdot 7}{5^{2}})$

Step 4.2.2.2

Raise $5$ to the power of $2$ .

$f (3) = \log (\frac{3 \cdot 7}{25})$

Step 4.2.3

Multiply $3$ by $7$ .

$f (3) = \log (\frac{21}{25})$

Step 4.2.4

The final answer is $\log (\frac{21}{25})$ .

$\log (\frac{21}{25})$

Step 4.3

Convert $\log (\frac{21}{25})$ to decimal.

$y = - 0.07572071$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0, x = - 4$ and the points $(1, - 0.2552725), (2, - 0.12493873), (3, - 0.07572071)$ .

Vertical Asymptote: $x = 0, x = - 4$

$\begin{array}{c} x & y \\ 1 & - 0.255 \\ 2 & - 0.125 \\ 3 & - 0.076 \end{array}$

Step 6