Calculus Examples

$\ln (x^{2} + 9)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x^{2} + 9 = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Subtract $9$ from both sides of the equation.

$x^{2} = - 9$

Step 1.2.2

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{- 9}$

Step 1.2.3

Simplify $\pm \sqrt{- 9}$ .

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Step 1.2.3.1

Rewrite $- 9$ as $- 1 (9)$ .

$x = \pm \sqrt{- 1 (9)}$

Step 1.2.3.2

Rewrite $\sqrt{- 1 (9)}$ as $\sqrt{- 1} \cdot \sqrt{9}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{9}$

Step 1.2.3.3

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

$x = \pm i \cdot \sqrt{9}$

Step 1.2.3.4

Rewrite $9$ as $3^{2}$ .

$x = \pm i \cdot \sqrt{3^{2}}$

Step 1.2.3.5

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm i \cdot 3$

Step 1.2.3.6

Move $3$ to the left of $i$ .

$x = \pm 3 i$

Step 1.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = 3 i$

Step 1.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - 3 i$

Step 1.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = 3 i, - 3 i$

Step 1.3

The vertical asymptote occurs at $x = 3 i, x = - 3 i$ .

Vertical Asymptote: $x = 3 i, x = - 3 i$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = \ln ({(1)}^{2} + 9)$

Step 2.2

Simplify the result.

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Step 2.2.1

One to any power is one.

$f (1) = \ln (1 + 9)$

Step 2.2.2

Add $1$ and $9$ .

$f (1) = \ln (10)$

Step 2.2.3

The final answer is $\ln (10)$ .

$\ln (10)$

Step 2.3

Convert $\ln (10)$ to decimal.

$y = 2.30258509$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = \ln ({(2)}^{2} + 9)$

Step 3.2

Simplify the result.

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Step 3.2.1

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

$f (2) = \ln (4 + 9)$

Step 3.2.2

Add $4$ and $9$ .

$f (2) = \ln (13)$

Step 3.2.3

The final answer is $\ln (13)$ .

$\ln (13)$

Step 3.3

Convert $\ln (13)$ to decimal.

$y = 2.56494935$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = \ln ({(3)}^{2} + 9)$

Step 4.2

Simplify the result.

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Step 4.2.1

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

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

Step 4.2.2

Add $9$ and $9$ .

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

Step 4.2.3

The final answer is $\ln (18)$ .

$\ln (18)$

Step 4.3

Convert $\ln (18)$ to decimal.

$y = 2.89037175$

Step 5

The log function can be graphed using the vertical asymptote at $x = 3 i, x = - 3 i$ and the points $(1, 2.30258509), (2, 2.56494935), (3, 2.89037175)$ .

Vertical Asymptote: $x = 3 i, x = - 3 i$

$\begin{array}{c} x & y \\ 1 & 2.303 \\ 2 & 2.565 \\ 3 & 2.89 \end{array}$

Step 6