Calculus Examples

$\ln (x^{2} + 3 x + 7)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x^{2} + 3 x + 7 = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 1.2.2

Substitute the values $a = 1$ , $b = 3$ , and $c = 7$ into the quadratic formula and solve for $x$ .

$\frac{- 3 \pm \sqrt{3^{2} - 4 \cdot (1 \cdot 7)}}{2 \cdot 1}$

Step 1.2.3

Simplify.

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Step 1.2.3.1

Simplify the numerator.

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Step 1.2.3.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Step 1.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 7$ .

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Step 1.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 7}}{2 \cdot 1}$

Step 1.2.3.1.2.2

Multiply $- 4$ by $7$ .

$x = \frac{- 3 \pm \sqrt{9 - 28}}{2 \cdot 1}$

Step 1.2.3.1.3

Subtract $28$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 19}}{2 \cdot 1}$

Step 1.2.3.1.4

Rewrite $- 19$ as $- 1 (19)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 19}}{2 \cdot 1}$

Step 1.2.3.1.5

Rewrite $\sqrt{- 1 (19)}$ as $\sqrt{- 1} \cdot \sqrt{19}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{19}}{2 \cdot 1}$

Step 1.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \sqrt{19}}{2 \cdot 1}$

Step 1.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{19}}{2}$

Step 1.2.4

Simplify the expression to solve for the $+$ portion of the $\pm$ .

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Step 1.2.4.1

Simplify the numerator.

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Step 1.2.4.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Step 1.2.4.1.2

Multiply $- 4 \cdot 1 \cdot 7$ .

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Step 1.2.4.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 7}}{2 \cdot 1}$

Step 1.2.4.1.2.2

Multiply $- 4$ by $7$ .

$x = \frac{- 3 \pm \sqrt{9 - 28}}{2 \cdot 1}$

Step 1.2.4.1.3

Subtract $28$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 19}}{2 \cdot 1}$

Step 1.2.4.1.4

Rewrite $- 19$ as $- 1 (19)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 19}}{2 \cdot 1}$

Step 1.2.4.1.5

Rewrite $\sqrt{- 1 (19)}$ as $\sqrt{- 1} \cdot \sqrt{19}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{19}}{2 \cdot 1}$

Step 1.2.4.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \sqrt{19}}{2 \cdot 1}$

Step 1.2.4.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{19}}{2}$

Step 1.2.4.3

Change the $\pm$ to $+$ .

$x = \frac{- 3 + i \sqrt{19}}{2}$

Step 1.2.4.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 + i \sqrt{19}}{2}$

Step 1.2.4.5

Factor $- 1$ out of $i \sqrt{19}$ .

$x = \frac{- 1 \cdot 3 - (- i \sqrt{19})}{2}$

Step 1.2.4.6

Factor $- 1$ out of $- 1 (3) - (- i \sqrt{19})$ .

$x = \frac{- 1 (3 - i \sqrt{19})}{2}$

Step 1.2.4.7

Move the negative in front of the fraction.

$x = - \frac{3 - i \sqrt{19}}{2}$

Step 1.2.5

Simplify the expression to solve for the $-$ portion of the $\pm$ .

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

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

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Step 1.2.5.1.2

Multiply $- 4 \cdot 1 \cdot 7$ .

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Step 1.2.5.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 3 \pm \sqrt{9 - 4 \cdot 7}}{2 \cdot 1}$

Step 1.2.5.1.2.2

Multiply $- 4$ by $7$ .

$x = \frac{- 3 \pm \sqrt{9 - 28}}{2 \cdot 1}$

Step 1.2.5.1.3

Subtract $28$ from $9$ .

$x = \frac{- 3 \pm \sqrt{- 19}}{2 \cdot 1}$

Step 1.2.5.1.4

Rewrite $- 19$ as $- 1 (19)$ .

$x = \frac{- 3 \pm \sqrt{- 1 \cdot 19}}{2 \cdot 1}$

Step 1.2.5.1.5

Rewrite $\sqrt{- 1 (19)}$ as $\sqrt{- 1} \cdot \sqrt{19}$ .

$x = \frac{- 3 \pm \sqrt{- 1} \cdot \sqrt{19}}{2 \cdot 1}$

Step 1.2.5.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 3 \pm i \sqrt{19}}{2 \cdot 1}$

Step 1.2.5.2

Multiply $2$ by $1$ .

$x = \frac{- 3 \pm i \sqrt{19}}{2}$

Step 1.2.5.3

Change the $\pm$ to $-$ .

$x = \frac{- 3 - i \sqrt{19}}{2}$

Step 1.2.5.4

Rewrite $- 3$ as $- 1 (3)$ .

$x = \frac{- 1 \cdot 3 - i \sqrt{19}}{2}$

Step 1.2.5.5

Factor $- 1$ out of $- i \sqrt{19}$ .

$x = \frac{- 1 \cdot 3 - (i \sqrt{19})}{2}$

Step 1.2.5.6

Factor $- 1$ out of $- 1 (3) - (i \sqrt{19})$ .

$x = \frac{- 1 (3 + i \sqrt{19})}{2}$

Step 1.2.5.7

Move the negative in front of the fraction.

$x = - \frac{3 + i \sqrt{19}}{2}$

Step 1.2.6

The final answer is the combination of both solutions.

$x = - \frac{3 - i \sqrt{19}}{2}, - \frac{3 + i \sqrt{19}}{2}$

Step 1.3

The vertical asymptote occurs at $x = - \frac{3 - i \sqrt{19}}{2}, x = - \frac{3 + i \sqrt{19}}{2}$ .

Vertical Asymptote: $x = - \frac{3 - i \sqrt{19}}{2}, x = - \frac{3 + i \sqrt{19}}{2}$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = \ln ({(1)}^{2} + 3 (1) + 7)$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

One to any power is one.

$f (1) = \ln (1 + 3 (1) + 7)$

Step 2.2.1.2

Multiply $3$ by $1$ .

$f (1) = \ln (1 + 3 + 7)$

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add $1$ and $3$ .

$f (1) = \ln (4 + 7)$

Step 2.2.2.2

Add $4$ and $7$ .

$f (1) = \ln (11)$

Step 2.2.3

The final answer is $\ln (11)$ .

$\ln (11)$

Step 2.3

Convert $\ln (11)$ to decimal.

$y = 2.39789527$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = \ln ({(2)}^{2} + 3 (2) + 7)$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

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

$f (2) = \ln (4 + 3 (2) + 7)$

Step 3.2.1.2

Multiply $3$ by $2$ .

$f (2) = \ln (4 + 6 + 7)$

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add $4$ and $6$ .

$f (2) = \ln (10 + 7)$

Step 3.2.2.2

Add $10$ and $7$ .

$f (2) = \ln (17)$

Step 3.2.3

The final answer is $\ln (17)$ .

$\ln (17)$

Step 3.3

Convert $\ln (17)$ to decimal.

$y = 2.83321334$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = \ln ({(3)}^{2} + 3 (3) + 7)$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

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

$f (3) = \ln (9 + 3 (3) + 7)$

Step 4.2.1.2

Multiply $3$ by $3$ .

$f (3) = \ln (9 + 9 + 7)$

Step 4.2.2

Simplify by adding numbers.

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Step 4.2.2.1

Add $9$ and $9$ .

$f (3) = \ln (18 + 7)$

Step 4.2.2.2

Add $18$ and $7$ .

$f (3) = \ln (25)$

Step 4.2.3

The final answer is $\ln (25)$ .

$\ln (25)$

Step 4.3

Convert $\ln (25)$ to decimal.

$y = 3.21887582$

Step 5

The log function can be graphed using the vertical asymptote at $x = - \frac{3 - i \sqrt{19}}{2}, x = - \frac{3 + i \sqrt{19}}{2}$ and the points $(1, 2.39789527), (2, 2.83321334), (3, 3.21887582)$ .

Vertical Asymptote: $x = - \frac{3 - i \sqrt{19}}{2}, x = - \frac{3 + i \sqrt{19}}{2}$

$\begin{array}{c} x & y \\ 1 & 2.398 \\ 2 & 2.833 \\ 3 & 3.219 \end{array}$

Step 6