Calculus Examples

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

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

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

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

Step 1.2

Solve for

x

$x$ .

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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}

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

Step 1.2.2

Substitute the values

a = 1

$a = 1$ ,

b = 3

$b = 3$ , and

c = 7

$c = 7$ into the quadratic formula and solve for

x

$x$ .

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

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

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

$3$ to the power of

2

$2$ .

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

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

Step 1.2.3.1.2

Multiply

- 4 \cdot 1 \cdot 7

$- 4 \cdot 1 \cdot 7$ .

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Step 1.2.3.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

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

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

Step 1.2.3.1.2.2

Multiply

- 4

$- 4$ by

7

$7$ .

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

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

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

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

Step 1.2.3.1.3

Subtract

28

$28$ from

9

$9$ .

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

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

Step 1.2.3.1.4

Rewrite

- 19

$- 19$ as

- 1 (19)

$- 1 (19)$ .

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

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

Step 1.2.3.1.5

Rewrite

\sqrt{- 1 (19)}

$\sqrt{- 1 (19)}$ as

\sqrt{- 1} \cdot \sqrt{19}

$\sqrt{- 1} \cdot \sqrt{19}$ .

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

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

Step 1.2.3.1.6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

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

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

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

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

Step 1.2.3.2

Multiply

2

$2$ by

1

$1$ .

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

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

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

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

Step 1.2.4

Simplify the expression to solve for the

+

$+$ portion of the

\pm

$\pm$ .

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Step 1.2.4.1

Simplify the numerator.

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Step 1.2.4.1.1

Raise

3

$3$ to the power of

2

$2$ .

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

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

Step 1.2.4.1.2

Multiply

- 4 \cdot 1 \cdot 7

$- 4 \cdot 1 \cdot 7$ .

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Step 1.2.4.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

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

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

Step 1.2.4.1.2.2

Multiply

- 4

$- 4$ by

7

$7$ .

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

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

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

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

Step 1.2.4.1.3

Subtract

28

$28$ from

9

$9$ .

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

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

Step 1.2.4.1.4

Rewrite

- 19

$- 19$ as

- 1 (19)

$- 1 (19)$ .

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

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

Step 1.2.4.1.5

Rewrite

\sqrt{- 1 (19)}

$\sqrt{- 1 (19)}$ as

\sqrt{- 1} \cdot \sqrt{19}

$\sqrt{- 1} \cdot \sqrt{19}$ .

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

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

Step 1.2.4.1.6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

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

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

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

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

Step 1.2.4.2

Multiply

2

$2$ by

1

$1$ .

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

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

Step 1.2.4.3

Change the

\pm

$\pm$ to

+

$+$ .

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

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

Step 1.2.4.4

Rewrite

- 3

$- 3$ as

- 1 (3)

$- 1 (3)$ .

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

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

Step 1.2.4.5

Factor

- 1

$- 1$ out of

i \sqrt{19}

$i \sqrt{19}$ .

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

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

Step 1.2.4.6

Factor

- 1

$- 1$ out of

- 1 (3) - (- i \sqrt{19})

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

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

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

Step 1.2.4.7

Move the negative in front of the fraction.

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

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

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

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

Step 1.2.5

Simplify the expression to solve for the

-

$-$ portion of the

\pm

$\pm$ .

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Step 1.2.5.1

Simplify the numerator.

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Step 1.2.5.1.1

Raise

3

$3$ to the power of

2

$2$ .

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

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

Step 1.2.5.1.2

Multiply

- 4 \cdot 1 \cdot 7

$- 4 \cdot 1 \cdot 7$ .

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Step 1.2.5.1.2.1

Multiply

- 4

$- 4$ by

1

$1$ .

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

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

Step 1.2.5.1.2.2

Multiply

- 4

$- 4$ by

7

$7$ .

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

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

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

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

Step 1.2.5.1.3

Subtract

28

$28$ from

9

$9$ .

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

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

Step 1.2.5.1.4

Rewrite

- 19

$- 19$ as

- 1 (19)

$- 1 (19)$ .

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

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

Step 1.2.5.1.5

Rewrite

\sqrt{- 1 (19)}

$\sqrt{- 1 (19)}$ as

\sqrt{- 1} \cdot \sqrt{19}

$\sqrt{- 1} \cdot \sqrt{19}$ .

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

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

Step 1.2.5.1.6

Rewrite

\sqrt{- 1}

$\sqrt{- 1}$ as

i

$i$ .

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

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

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

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

Step 1.2.5.2

Multiply

2

$2$ by

1

$1$ .

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

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

Step 1.2.5.3

Change the

\pm

$\pm$ to

-

$-$ .

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

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

Step 1.2.5.4

Rewrite

- 3

$- 3$ as

- 1 (3)

$- 1 (3)$ .

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

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

Step 1.2.5.5

Factor

- 1

$- 1$ out of

- i \sqrt{19}

$- i \sqrt{19}$ .

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

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

Step 1.2.5.6

Factor

- 1

$- 1$ out of

- 1 (3) - (i \sqrt{19})

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

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

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

Step 1.2.5.7

Move the negative in front of the fraction.

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

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

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

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

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}

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

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

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

Step 1.3

The vertical asymptote occurs at

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

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

Vertical Asymptote:

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

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

Vertical Asymptote:

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

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

Step 2

Find the point at

x = 1

$x = 1$ .

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Step 2.1

Replace the variable

x

$x$ with

1

$1$ in the expression.

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

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

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)

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

Step 2.2.1.2

Multiply

3

$3$ by

1

$1$ .

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

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

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

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

Step 2.2.2

Simplify by adding numbers.

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Step 2.2.2.1

Add

1

$1$ and

3

$3$ .

f (1) = ln (4 + 7)

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

Step 2.2.2.2

Add

4

$4$ and

7

$7$ .

f (1) = ln (11)

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

f (1) = ln (11)

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

Step 2.2.3

The final answer is

ln (11)

$\ln (11)$ .

ln (11)

$\ln (11)$

ln (11)

$\ln (11)$

Step 2.3

Convert

ln (11)

$\ln (11)$ to decimal.

y = 2.39789527

$y = 2.39789527$

y = 2.39789527

$y = 2.39789527$

Step 3

Find the point at

x = 2

$x = 2$ .

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Step 3.1

Replace the variable

x

$x$ with

2

$2$ in the expression.

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

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

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

$2$ to the power of

2

$2$ .

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

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

Step 3.2.1.2

Multiply

3

$3$ by

2

$2$ .

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

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

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

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

Step 3.2.2

Simplify by adding numbers.

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Step 3.2.2.1

Add

4

$4$ and

6

$6$ .

f (2) = ln (10 + 7)

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

Step 3.2.2.2

Add

10

$10$ and

7

$7$ .

f (2) = ln (17)

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

f (2) = ln (17)

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

Step 3.2.3

The final answer is

ln (17)

$\ln (17)$ .

ln (17)

$\ln (17)$

ln (17)

$\ln (17)$

Step 3.3

Convert

ln (17)

$\ln (17)$ to decimal.

y = 2.83321334

$y = 2.83321334$

y = 2.83321334

$y = 2.83321334$

Step 4

Find the point at

x = 3

$x = 3$ .

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Step 4.1

Replace the variable

x

$x$ with

3

$3$ in the expression.

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

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

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

$3$ to the power of

2

$2$ .

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

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

Step 4.2.1.2

Multiply

3

$3$ by

3

$3$ .

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

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

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

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

Step 4.2.2

Simplify by adding numbers.

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Step 4.2.2.1

Add

9

$9$ and

9

$9$ .

f (3) = ln (18 + 7)

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

Step 4.2.2.2

Add

18

$18$ and

7

$7$ .

f (3) = ln (25)

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

f (3) = ln (25)

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

Step 4.2.3

The final answer is

ln (25)

$\ln (25)$ .

ln (25)

$\ln (25)$

ln (25)

$\ln (25)$

Step 4.3

Convert

ln (25)

$\ln (25)$ to decimal.

y = 3.21887582

$y = 3.21887582$

y = 3.21887582

$y = 3.21887582$

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}

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

(1, 2.39789527), (2, 2.83321334), (3, 3.21887582)

$(1, 2.39789527), (2, 2.83321334), (3, 3.21887582)$ .

Vertical Asymptote:

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

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

\begin{matrix} x & y 1 & 2.398 2 & 2.833 3 & 3.219 \end{matrix}

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

Step 6