Calculus Examples

$f (x) = \ln (x^{2} + 64)$

Step 1

Set the argument of the logarithm equal to zero.

$x^{2} + 64 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $64$ from both sides of the equation.

$x^{2} = - 64$

Step 2.2

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

$x = \pm \sqrt{- 64}$

Step 2.3

Simplify $\pm \sqrt{- 64}$ .

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Step 2.3.1

Rewrite $- 64$ as $- 1 (64)$ .

$x = \pm \sqrt{- 1 (64)}$

Step 2.3.2

Rewrite $\sqrt{- 1 (64)}$ as $\sqrt{- 1} \cdot \sqrt{64}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{64}$

Step 2.3.3

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

$x = \pm i \cdot \sqrt{64}$

Step 2.3.4

Rewrite $64$ as $8^{2}$ .

$x = \pm i \cdot \sqrt{8^{2}}$

Step 2.3.5

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

$x = \pm i \cdot 8$

Step 2.3.6

Move $8$ to the left of $i$ .

$x = \pm 8 i$

Step 2.4

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

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Step 2.4.1

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

$x = 8 i$

Step 2.4.2

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

$x = - 8 i$

Step 2.4.3

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

$x = 8 i, - 8 i$

Step 3

The vertical asymptote occurs at $x = 8 i, x = - 8 i$ .

Vertical Asymptote: $x = 8 i, x = - 8 i$

Step 4