Calculus Examples

$f (x) = \ln (x^{2} + 25)$

Step 1

Find the x-intercepts.

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Step 1.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = \ln (x^{2} + 25)$

Step 1.2

Solve the equation.

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Step 1.2.1

Rewrite the equation as $\ln (x^{2} + 25) = 0$ .

$\ln (x^{2} + 25) = 0$

Step 1.2.2

To solve for $x$ , rewrite the equation using properties of logarithms.

$e^{\ln (x^{2} + 25)} = e^{0}$

Step 1.2.3

Rewrite $\ln (x^{2} + 25) = 0$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{0} = x^{2} + 25$

Step 1.2.4

Solve for $x$ .

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Step 1.2.4.1

Rewrite the equation as $x^{2} + 25 = e^{0}$ .

$x^{2} + 25 = e^{0}$

Step 1.2.4.2

Anything raised to $0$ is $1$ .

$x^{2} + 25 = 1$

Step 1.2.4.3

Move all terms not containing $x$ to the right side of the equation.

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Step 1.2.4.3.1

Subtract $25$ from both sides of the equation.

$x^{2} = 1 - 25$

Step 1.2.4.3.2

Subtract $25$ from $1$ .

$x^{2} = - 24$

Step 1.2.4.4

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

$x = \pm \sqrt{- 24}$

Step 1.2.4.5

Simplify $\pm \sqrt{- 24}$ .

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Step 1.2.4.5.1

Rewrite $- 24$ as $- 1 (24)$ .

$x = \pm \sqrt{- 1 (24)}$

Step 1.2.4.5.2

Rewrite $\sqrt{- 1 (24)}$ as $\sqrt{- 1} \cdot \sqrt{24}$ .

$x = \pm \sqrt{- 1} \cdot \sqrt{24}$

Step 1.2.4.5.3

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

$x = \pm i \cdot \sqrt{24}$

Step 1.2.4.5.4

Rewrite $24$ as $2^{2} \cdot 6$ .

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Step 1.2.4.5.4.1

Factor $4$ out of $24$ .

$x = \pm i \cdot \sqrt{4 (6)}$

Step 1.2.4.5.4.2

Rewrite $4$ as $2^{2}$ .

$x = \pm i \cdot \sqrt{2^{2} \cdot 6}$

Step 1.2.4.5.5

Pull terms out from under the radical.

$x = \pm i \cdot (2 \sqrt{6})$

Step 1.2.4.5.6

Move $2$ to the left of $i$ .

$x = \pm 2 i \sqrt{6}$

Step 1.2.4.6

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

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Step 1.2.4.6.1

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

$x = 2 i \sqrt{6}$

Step 1.2.4.6.2

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

$x = - 2 i \sqrt{6}$

Step 1.2.4.6.3

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

$x = 2 i \sqrt{6}, - 2 i \sqrt{6}$

Step 1.3

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

x-intercept(s): $None$

Step 2

Find the y-intercepts.

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Step 2.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = \ln ({(0)}^{2} + 25)$

Step 2.2

Solve the equation.

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Step 2.2.1

Remove parentheses.

$y = \ln (0^{2} + 25)$

Step 2.2.2

Remove parentheses.

$y = \ln ({(0)}^{2} + 25)$

Step 2.2.3

Simplify $\ln ({(0)}^{2} + 25)$ .

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Step 2.2.3.1

Raising $0$ to any positive power yields $0$ .

$y = \ln (0 + 25)$

Step 2.2.3.2

Add $0$ and $25$ .

$y = \ln (25)$

Step 2.3

y-intercept(s) in point form.

y-intercept(s): $(0, \ln (25))$

Step 3

List the intersections.

x-intercept(s): $None$

y-intercept(s): $(0, \ln (25))$

Step 4