Calculus Examples

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

Step 1

Find the asymptotes.

Tap for more steps...

Step 1.1

Set the argument of the logarithm equal to zero.

$x^{2} + 64 = 0$

Step 1.2

Solve for $x$ .

Tap for more steps...

Step 1.2.1

Subtract $64$ from both sides of the equation.

$x^{2} = - 64$

Step 1.2.2

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

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

Step 1.2.3

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

Tap for more steps...

Step 1.2.3.1

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

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

Step 1.2.3.2

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

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

Step 1.2.3.3

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

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

Step 1.2.3.4

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

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

Step 1.2.3.5

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

$x = \pm i \cdot 8$

Step 1.2.3.6

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

$x = \pm 8 i$

Step 1.2.4

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

Tap for more steps...

Step 1.2.4.1

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

$x = 8 i$

Step 1.2.4.2

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

$x = - 8 i$

Step 1.2.4.3

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

$x = 8 i, - 8 i$

Step 1.3

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

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

Step 2

Find the point at $x = 1$ .

Tap for more steps...

Step 2.1

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

$f (1) = \ln ({(1)}^{2} + 64)$

Step 2.2

Simplify the result.

Tap for more steps...

Step 2.2.1

One to any power is one.

$f (1) = \ln (1 + 64)$

Step 2.2.2

Add $1$ and $64$ .

$f (1) = \ln (65)$

Step 2.2.3

The final answer is $\ln (65)$ .

$\ln (65)$

Step 2.3

Convert $\ln (65)$ to decimal.

$y = 4.17438726$

Step 3

Find the point at $x = 2$ .

Tap for more steps...

Step 3.1

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

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

Step 3.2

Simplify the result.

Tap for more steps...

Step 3.2.1

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

$f (2) = \ln (4 + 64)$

Step 3.2.2

Add $4$ and $64$ .

$f (2) = \ln (68)$

Step 3.2.3

The final answer is $\ln (68)$ .

$\ln (68)$

Step 3.3

Convert $\ln (68)$ to decimal.

$y = 4.2195077$

Step 4

Find the point at $x = 3$ .

Tap for more steps...

Step 4.1

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

$f (3) = \ln ({(3)}^{2} + 64)$

Step 4.2

Simplify the result.

Tap for more steps...

Step 4.2.1

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

$f (3) = \ln (9 + 64)$

Step 4.2.2

Add $9$ and $64$ .

$f (3) = \ln (73)$

Step 4.2.3

The final answer is $\ln (73)$ .

$\ln (73)$

Step 4.3

Convert $\ln (73)$ to decimal.

$y = 4.29045944$

Step 5

The log function can be graphed using the vertical asymptote at $x = 8 i, x = - 8 i$ and the points $(1, 4.17438726), (2, 4.2195077), (3, 4.29045944)$ .

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

$\begin{array}{c} x & y \\ 1 & 4.174 \\ 2 & 4.22 \\ 3 & 4.29 \end{array}$

Step 6