Trigonometry Examples

$f (x) = \log (14 - x) + \log (x - 5)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$19 x - 70 - x^{2} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Factor the left side of the equation.

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Step 1.2.1.1

Factor $- 1$ out of $19 x - 70 - x^{2}$ .

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Step 1.2.1.1.1

Reorder the expression.

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Step 1.2.1.1.1.1

Move $- 70$ .

$19 x - x^{2} - 70 = 0$

Step 1.2.1.1.1.2

Reorder $19 x$ and $- x^{2}$ .

$- x^{2} + 19 x - 70 = 0$

Step 1.2.1.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 19 x - 70 = 0$

Step 1.2.1.1.3

Factor $- 1$ out of $19 x$ .

$- (x^{2}) - (- 19 x) - 70 = 0$

Step 1.2.1.1.4

Rewrite $- 70$ as $- 1 (70)$ .

$- (x^{2}) - (- 19 x) - 1 \cdot 70 = 0$

Step 1.2.1.1.5

Factor $- 1$ out of $- (x^{2}) - (- 19 x)$ .

$- (x^{2} - 19 x) - 1 \cdot 70 = 0$

Step 1.2.1.1.6

Factor $- 1$ out of $- (x^{2} - 19 x) - 1 (70)$ .

$- (x^{2} - 19 x + 70) = 0$

Step 1.2.1.2

Factor.

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Step 1.2.1.2.1

Factor $x^{2} - 19 x + 70$ using the AC method.

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Step 1.2.1.2.1.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $70$ and whose sum is $- 19$ .

$- 14, - 5$

Step 1.2.1.2.1.2

Write the factored form using these integers.

$- ((x - 14) (x - 5)) = 0$

Step 1.2.1.2.2

Remove unnecessary parentheses.

$- (x - 14) (x - 5) = 0$

Step 1.2.2

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

$x - 14 = 0$

$x - 5 = 0$

Step 1.2.3

Set $x - 14$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $x - 14$ equal to $0$ .

$x - 14 = 0$

Step 1.2.3.2

Add $14$ to both sides of the equation.

$x = 14$

Step 1.2.4

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 1.2.4.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 1.2.4.2

Add $5$ to both sides of the equation.

$x = 5$

Step 1.2.5

The final solution is all the values that make $- (x - 14) (x - 5) = 0$ true.

$x = 14, 5$

Step 1.3

The vertical asymptote occurs at $x = 14, x = 5$ .

Vertical Asymptote: $x = 14, x = 5$

Step 2

The log function can be graphed using the vertical asymptote at $x = 14, x = 5$ and the points .

Vertical Asymptote: $x = 14, x = 5$

$\begin{array}{c} x & y \end{array}$

Step 3