Calculus Examples

2 ln (y)

$2 \ln (y)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

y^{2} = 0

$y^{2} = 0$

Step 1.2

Solve for

x

$x$ .

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Step 1.2.1

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

y = \pm \sqrt{0}

$y = \pm \sqrt{0}$

Step 1.2.2

Simplify

\pm \sqrt{0}

$\pm \sqrt{0}$ .

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Step 1.2.2.1

Rewrite

0

$0$ as

0^{2}

$0^{2}$ .

y = \pm \sqrt{0^{2}}

$y = \pm \sqrt{0^{2}}$

Step 1.2.2.2

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

y = \pm 0

$y = \pm 0$

Step 1.2.2.3

Plus or minus

0

$0$ is

0

$0$ .

y = 0

$y = 0$

y = 0

$y = 0$

y = 0

$y = 0$

Step 1.3

The vertical asymptote occurs at

y = 0

$y = 0$ .

Vertical Asymptote:

y = 0

$y = 0$

Vertical Asymptote:

y = 0

$y = 0$

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) = \sqrt{e^{1}}, - \sqrt{e^{1}}

$f (1) = \sqrt{e^{1}}, - \sqrt{e^{1}}$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify.

f (1) = \sqrt{e}, - \sqrt{e}

$f (1) = \sqrt{e}, - \sqrt{e}$

Step 2.2.2

The final answer is

\sqrt{e}, - \sqrt{e}

$\sqrt{e}, - \sqrt{e}$ .

\sqrt{e}, - \sqrt{e}

$\sqrt{e}, - \sqrt{e}$

\sqrt{e}, - \sqrt{e}

$\sqrt{e}, - \sqrt{e}$

Step 2.3

Convert

\sqrt{e}, - \sqrt{e}

$\sqrt{e}, - \sqrt{e}$ to decimal.

y = \sqrt{e}, - \sqrt{e}

$y = \sqrt{e}, - \sqrt{e}$

y = \sqrt{e}, - \sqrt{e}

$y = \sqrt{e}, - \sqrt{e}$

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) = \sqrt{e^{2}}, - \sqrt{e^{2}}

$f (2) = \sqrt{e^{2}}, - \sqrt{e^{2}}$

Step 3.2

Simplify the result.

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Step 3.2.1

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

f (2) = e, - \sqrt{e^{2}}

$f (2) = e, - \sqrt{e^{2}}$

Step 3.2.2

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

f (2) = e, - e

$f (2) = e, - e$

Step 3.2.3

The final answer is

e, - e

$e, - e$ .

e, - e

$e, - e$

e, - e

$e, - e$

Step 3.3

Convert

e, - e

$e, - e$ to decimal.

y = e, - e

$y = e, - e$

y = e, - e

$y = e, - e$

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) = \sqrt{e^{3}}, - \sqrt{e^{3}}

$f (3) = \sqrt{e^{3}}, - \sqrt{e^{3}}$

Step 4.2

Simplify the result.

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Step 4.2.1

Factor out

e^{2}

$e^{2}$ .

f (3) = \sqrt{e^{2} e}, - \sqrt{e^{3}}

$f (3) = \sqrt{e^{2} e}, - \sqrt{e^{3}}$

Step 4.2.2

Pull terms out from under the radical.

f (3) = e \sqrt{e}, - \sqrt{e^{3}}

$f (3) = e \sqrt{e}, - \sqrt{e^{3}}$

Step 4.2.3

Factor out

e^{2}

$e^{2}$ .

f (3) = e \sqrt{e}, - \sqrt{e^{2} e}

$f (3) = e \sqrt{e}, - \sqrt{e^{2} e}$

Step 4.2.4

Pull terms out from under the radical.

f (3) = e \sqrt{e}, - e \sqrt{e}

$f (3) = e \sqrt{e}, - e \sqrt{e}$

Step 4.2.5

The final answer is

e \sqrt{e}, - e \sqrt{e}

$e \sqrt{e}, - e \sqrt{e}$ .

e \sqrt{e}, - e \sqrt{e}

$e \sqrt{e}, - e \sqrt{e}$

e \sqrt{e}, - e \sqrt{e}

$e \sqrt{e}, - e \sqrt{e}$

Step 4.3

Convert

e \sqrt{e}, - e \sqrt{e}

$e \sqrt{e}, - e \sqrt{e}$ to decimal.

y = e \sqrt{e}, - e \sqrt{e}

$y = e \sqrt{e}, - e \sqrt{e}$

y = e \sqrt{e}, - e \sqrt{e}

$y = e \sqrt{e}, - e \sqrt{e}$

Step 5

Find the point at

x = 4

$x = 4$ .

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Step 5.1

Replace the variable

x

$x$ with

4

$4$ in the expression.

f (4) = \sqrt{e^{4}}, - \sqrt{e^{4}}

$f (4) = \sqrt{e^{4}}, - \sqrt{e^{4}}$

Step 5.2

Simplify the result.

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Step 5.2.1

Rewrite

e^{4}

$e^{4}$ as

{(e^{2})}^{2}

${(e^{2})}^{2}$ .

f (4) = \sqrt{{(e^{2})}^{2}}, - \sqrt{e^{4}}

$f (4) = \sqrt{{(e^{2})}^{2}}, - \sqrt{e^{4}}$

Step 5.2.2

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

f (4) = e^{2}, - \sqrt{e^{4}}

$f (4) = e^{2}, - \sqrt{e^{4}}$

Step 5.2.3

Rewrite

e^{4}

$e^{4}$ as

{(e^{2})}^{2}

${(e^{2})}^{2}$ .

f (4) = e^{2}, - \sqrt{{(e^{2})}^{2}}

$f (4) = e^{2}, - \sqrt{{(e^{2})}^{2}}$

Step 5.2.4

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

f (4) = e^{2}, - e^{2}

$f (4) = e^{2}, - e^{2}$

Step 5.2.5

The final answer is

e^{2}, - e^{2}

$e^{2}, - e^{2}$ .

e^{2}, - e^{2}

$e^{2}, - e^{2}$

e^{2}, - e^{2}

$e^{2}, - e^{2}$

Step 5.3

Convert

e^{2}, - e^{2}

$e^{2}, - e^{2}$ to decimal.

y = e^{2}, - e^{2}

$y = e^{2}, - e^{2}$

y = e^{2}, - e^{2}

$y = e^{2}, - e^{2}$

Step 6

Find the point at

x = 5

$x = 5$ .

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Step 6.1

Replace the variable

x

$x$ with

5

$5$ in the expression.

f (5) = \sqrt{e^{5}}, - \sqrt{e^{5}}

$f (5) = \sqrt{e^{5}}, - \sqrt{e^{5}}$

Step 6.2

Simplify the result.

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Step 6.2.1

Rewrite

e^{5}

$e^{5}$ as

{(e^{2})}^{2} e

${(e^{2})}^{2} e$ .

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Step 6.2.1.1

Factor out

e^{4}

$e^{4}$ .

f (5) = \sqrt{e^{4} e}, - \sqrt{e^{5}}

$f (5) = \sqrt{e^{4} e}, - \sqrt{e^{5}}$

Step 6.2.1.2

Rewrite

e^{4}

$e^{4}$ as

{(e^{2})}^{2}

${(e^{2})}^{2}$ .

f (5) = \sqrt{{(e^{2})}^{2} e}, - \sqrt{e^{5}}

$f (5) = \sqrt{{(e^{2})}^{2} e}, - \sqrt{e^{5}}$

f (5) = \sqrt{{(e^{2})}^{2} e}, - \sqrt{e^{5}}

$f (5) = \sqrt{{(e^{2})}^{2} e}, - \sqrt{e^{5}}$

Step 6.2.2

Pull terms out from under the radical.

f (5) = e^{2} \sqrt{e}, - \sqrt{e^{5}}

$f (5) = e^{2} \sqrt{e}, - \sqrt{e^{5}}$

Step 6.2.3

Rewrite

e^{5}

$e^{5}$ as

{(e^{2})}^{2} e

${(e^{2})}^{2} e$ .

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Step 6.2.3.1

Factor out

e^{4}

$e^{4}$ .

f (5) = e^{2} \sqrt{e}, - \sqrt{e^{4} e}

$f (5) = e^{2} \sqrt{e}, - \sqrt{e^{4} e}$

Step 6.2.3.2

Rewrite

e^{4}

$e^{4}$ as

{(e^{2})}^{2}

${(e^{2})}^{2}$ .

f (5) = e^{2} \sqrt{e}, - \sqrt{{(e^{2})}^{2} e}

$f (5) = e^{2} \sqrt{e}, - \sqrt{{(e^{2})}^{2} e}$

f (5) = e^{2} \sqrt{e}, - \sqrt{{(e^{2})}^{2} e}

$f (5) = e^{2} \sqrt{e}, - \sqrt{{(e^{2})}^{2} e}$

Step 6.2.4

Pull terms out from under the radical.

f (5) = e^{2} \sqrt{e}, - e^{2} \sqrt{e}

$f (5) = e^{2} \sqrt{e}, - e^{2} \sqrt{e}$

Step 6.2.5

The final answer is

e^{2} \sqrt{e}, - e^{2} \sqrt{e}

$e^{2} \sqrt{e}, - e^{2} \sqrt{e}$ .

e^{2} \sqrt{e}, - e^{2} \sqrt{e}

$e^{2} \sqrt{e}, - e^{2} \sqrt{e}$

e^{2} \sqrt{e}, - e^{2} \sqrt{e}

$e^{2} \sqrt{e}, - e^{2} \sqrt{e}$

Step 6.3

Convert

e^{2} \sqrt{e}, - e^{2} \sqrt{e}

$e^{2} \sqrt{e}, - e^{2} \sqrt{e}$ to decimal.

y = e^{2} \sqrt{e}, - e^{2} \sqrt{e}

$y = e^{2} \sqrt{e}, - e^{2} \sqrt{e}$

y = e^{2} \sqrt{e}, - e^{2} \sqrt{e}

$y = e^{2} \sqrt{e}, - e^{2} \sqrt{e}$

Step 7

Find the point at

x = 6

$x = 6$ .

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Step 7.1

Replace the variable

x

$x$ with

6

$6$ in the expression.

f (6) = \sqrt{e^{6}}, - \sqrt{e^{6}}

$f (6) = \sqrt{e^{6}}, - \sqrt{e^{6}}$

Step 7.2

Simplify the result.

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Step 7.2.1

Rewrite

e^{6}

$e^{6}$ as

{(e^{3})}^{2}

${(e^{3})}^{2}$ .

f (6) = \sqrt{{(e^{3})}^{2}}, - \sqrt{e^{6}}

$f (6) = \sqrt{{(e^{3})}^{2}}, - \sqrt{e^{6}}$

Step 7.2.2

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

f (6) = e^{3}, - \sqrt{e^{6}}

$f (6) = e^{3}, - \sqrt{e^{6}}$

Step 7.2.3

Rewrite

e^{6}

$e^{6}$ as

{(e^{3})}^{2}

${(e^{3})}^{2}$ .

f (6) = e^{3}, - \sqrt{{(e^{3})}^{2}}

$f (6) = e^{3}, - \sqrt{{(e^{3})}^{2}}$

Step 7.2.4

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

f (6) = e^{3}, - e^{3}

$f (6) = e^{3}, - e^{3}$

Step 7.2.5

The final answer is

e^{3}, - e^{3}

$e^{3}, - e^{3}$ .

e^{3}, - e^{3}

$e^{3}, - e^{3}$

e^{3}, - e^{3}

$e^{3}, - e^{3}$

Step 7.3

Convert

e^{3}, - e^{3}

$e^{3}, - e^{3}$ to decimal.

y = e^{3}, - e^{3}

$y = e^{3}, - e^{3}$

y = e^{3}, - e^{3}

$y = e^{3}, - e^{3}$

Step 8

Find the point at

x = 7

$x = 7$ .

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Step 8.1

Replace the variable

x

$x$ with

7

$7$ in the expression.

f (7) = \sqrt{e^{7}}, - \sqrt{e^{7}}

$f (7) = \sqrt{e^{7}}, - \sqrt{e^{7}}$

Step 8.2

Simplify the result.

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Step 8.2.1

Rewrite

e^{7}

$e^{7}$ as

{(e^{3})}^{2} e

${(e^{3})}^{2} e$ .

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Step 8.2.1.1

Factor out

e^{6}

$e^{6}$ .

f (7) = \sqrt{e^{6} e}, - \sqrt{e^{7}}

$f (7) = \sqrt{e^{6} e}, - \sqrt{e^{7}}$

Step 8.2.1.2

Rewrite

e^{6}

$e^{6}$ as

{(e^{3})}^{2}

${(e^{3})}^{2}$ .

f (7) = \sqrt{{(e^{3})}^{2} e}, - \sqrt{e^{7}}

$f (7) = \sqrt{{(e^{3})}^{2} e}, - \sqrt{e^{7}}$

f (7) = \sqrt{{(e^{3})}^{2} e}, - \sqrt{e^{7}}

$f (7) = \sqrt{{(e^{3})}^{2} e}, - \sqrt{e^{7}}$

Step 8.2.2

Pull terms out from under the radical.

f (7) = e^{3} \sqrt{e}, - \sqrt{e^{7}}

$f (7) = e^{3} \sqrt{e}, - \sqrt{e^{7}}$

Step 8.2.3

Rewrite

e^{7}

$e^{7}$ as

{(e^{3})}^{2} e

${(e^{3})}^{2} e$ .

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Step 8.2.3.1

Factor out

e^{6}

$e^{6}$ .

f (7) = e^{3} \sqrt{e}, - \sqrt{e^{6} e}

$f (7) = e^{3} \sqrt{e}, - \sqrt{e^{6} e}$

Step 8.2.3.2

Rewrite

e^{6}

$e^{6}$ as

{(e^{3})}^{2}

${(e^{3})}^{2}$ .

f (7) = e^{3} \sqrt{e}, - \sqrt{{(e^{3})}^{2} e}

$f (7) = e^{3} \sqrt{e}, - \sqrt{{(e^{3})}^{2} e}$

f (7) = e^{3} \sqrt{e}, - \sqrt{{(e^{3})}^{2} e}

$f (7) = e^{3} \sqrt{e}, - \sqrt{{(e^{3})}^{2} e}$

Step 8.2.4

Pull terms out from under the radical.

f (7) = e^{3} \sqrt{e}, - e^{3} \sqrt{e}

$f (7) = e^{3} \sqrt{e}, - e^{3} \sqrt{e}$

Step 8.2.5

The final answer is

e^{3} \sqrt{e}, - e^{3} \sqrt{e}

$e^{3} \sqrt{e}, - e^{3} \sqrt{e}$ .

e^{3} \sqrt{e}, - e^{3} \sqrt{e}

$e^{3} \sqrt{e}, - e^{3} \sqrt{e}$

e^{3} \sqrt{e}, - e^{3} \sqrt{e}

$e^{3} \sqrt{e}, - e^{3} \sqrt{e}$

Step 8.3

Convert

e^{3} \sqrt{e}, - e^{3} \sqrt{e}

$e^{3} \sqrt{e}, - e^{3} \sqrt{e}$ to decimal.

y = e^{3} \sqrt{e}, - e^{3} \sqrt{e}

$y = e^{3} \sqrt{e}, - e^{3} \sqrt{e}$

y = e^{3} \sqrt{e}, - e^{3} \sqrt{e}

$y = e^{3} \sqrt{e}, - e^{3} \sqrt{e}$

Step 9

Find the point at

x = 8

$x = 8$ .

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Step 9.1

Replace the variable

x

$x$ with

8

$8$ in the expression.

f (8) = \sqrt{e^{8}}, - \sqrt{e^{8}}

$f (8) = \sqrt{e^{8}}, - \sqrt{e^{8}}$

Step 9.2

Simplify the result.

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Step 9.2.1

Rewrite

e^{8}

$e^{8}$ as

{(e^{4})}^{2}

${(e^{4})}^{2}$ .

f (8) = \sqrt{{(e^{4})}^{2}}, - \sqrt{e^{8}}

$f (8) = \sqrt{{(e^{4})}^{2}}, - \sqrt{e^{8}}$

Step 9.2.2

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

f (8) = e^{4}, - \sqrt{e^{8}}

$f (8) = e^{4}, - \sqrt{e^{8}}$

Step 9.2.3

Rewrite

e^{8}

$e^{8}$ as

{(e^{4})}^{2}

${(e^{4})}^{2}$ .

f (8) = e^{4}, - \sqrt{{(e^{4})}^{2}}

$f (8) = e^{4}, - \sqrt{{(e^{4})}^{2}}$

Step 9.2.4

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

f (8) = e^{4}, - e^{4}

$f (8) = e^{4}, - e^{4}$

Step 9.2.5

The final answer is

e^{4}, - e^{4}

$e^{4}, - e^{4}$ .

e^{4}, - e^{4}

$e^{4}, - e^{4}$

e^{4}, - e^{4}

$e^{4}, - e^{4}$

Step 9.3

Convert

e^{4}, - e^{4}

$e^{4}, - e^{4}$ to decimal.

y = e^{4}, - e^{4}

$y = e^{4}, - e^{4}$

y = e^{4}, - e^{4}

$y = e^{4}, - e^{4}$

Step 10

Find the point at

x = 9

$x = 9$ .

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Step 10.1

Replace the variable

x

$x$ with

9

$9$ in the expression.

f (9) = \sqrt{e^{9}}, - \sqrt{e^{9}}

$f (9) = \sqrt{e^{9}}, - \sqrt{e^{9}}$

Step 10.2

Simplify the result.

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Step 10.2.1

Rewrite

e^{9}

$e^{9}$ as

{(e^{4})}^{2} e

${(e^{4})}^{2} e$ .

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Step 10.2.1.1

Factor out

e^{8}

$e^{8}$ .

f (9) = \sqrt{e^{8} e}, - \sqrt{e^{9}}

$f (9) = \sqrt{e^{8} e}, - \sqrt{e^{9}}$

Step 10.2.1.2

Rewrite

e^{8}

$e^{8}$ as

{(e^{4})}^{2}

${(e^{4})}^{2}$ .

f (9) = \sqrt{{(e^{4})}^{2} e}, - \sqrt{e^{9}}

$f (9) = \sqrt{{(e^{4})}^{2} e}, - \sqrt{e^{9}}$

f (9) = \sqrt{{(e^{4})}^{2} e}, - \sqrt{e^{9}}

$f (9) = \sqrt{{(e^{4})}^{2} e}, - \sqrt{e^{9}}$

Step 10.2.2

Pull terms out from under the radical.

f (9) = e^{4} \sqrt{e}, - \sqrt{e^{9}}

$f (9) = e^{4} \sqrt{e}, - \sqrt{e^{9}}$

Step 10.2.3

Rewrite

e^{9}

$e^{9}$ as

{(e^{4})}^{2} e

${(e^{4})}^{2} e$ .

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Step 10.2.3.1

Factor out

e^{8}

$e^{8}$ .

f (9) = e^{4} \sqrt{e}, - \sqrt{e^{8} e}

$f (9) = e^{4} \sqrt{e}, - \sqrt{e^{8} e}$

Step 10.2.3.2

Rewrite

e^{8}

$e^{8}$ as

{(e^{4})}^{2}

${(e^{4})}^{2}$ .

f (9) = e^{4} \sqrt{e}, - \sqrt{{(e^{4})}^{2} e}

$f (9) = e^{4} \sqrt{e}, - \sqrt{{(e^{4})}^{2} e}$

f (9) = e^{4} \sqrt{e}, - \sqrt{{(e^{4})}^{2} e}

$f (9) = e^{4} \sqrt{e}, - \sqrt{{(e^{4})}^{2} e}$

Step 10.2.4

Pull terms out from under the radical.

f (9) = e^{4} \sqrt{e}, - e^{4} \sqrt{e}

$f (9) = e^{4} \sqrt{e}, - e^{4} \sqrt{e}$

Step 10.2.5

The final answer is

e^{4} \sqrt{e}, - e^{4} \sqrt{e}

$e^{4} \sqrt{e}, - e^{4} \sqrt{e}$ .

e^{4} \sqrt{e}, - e^{4} \sqrt{e}

$e^{4} \sqrt{e}, - e^{4} \sqrt{e}$

Step 10.3

Convert

$e^{4} \sqrt{e}, - e^{4} \sqrt{e}$ to decimal.

$y = e^{4} \sqrt{e}, - e^{4} \sqrt{e}$

Step 11

Find the point at

$x = 10$ .

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Step 11.1

Replace the variable

$x$ with

$10$ in the expression.

$f (10) = \sqrt{e^{10}}, - \sqrt{e^{10}}$

Step 11.2

Simplify the result.

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Step 11.2.1

Rewrite

$e^{10}$ as

${(e^{5})}^{2}$ .

$f (10) = \sqrt{{(e^{5})}^{2}}, - \sqrt{e^{10}}$

Step 11.2.2

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

$f (10) = e^{5}, - \sqrt{e^{10}}$

Step 11.2.3

Rewrite

$e^{10}$ as

${(e^{5})}^{2}$ .

$f (10) = e^{5}, - \sqrt{{(e^{5})}^{2}}$

Step 11.2.4

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

$f (10) = e^{5}, - e^{5}$

Step 11.2.5

The final answer is

$e^{5}, - e^{5}$ .

$e^{5}, - e^{5}$

Step 11.3

Convert

$e^{5}, - e^{5}$ to decimal.

$y = e^{5}, - e^{5}$

Step 12

Find the point at

$x = 1$ .

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Step 12.1

Replace the variable

$x$ with

$1$ in the expression.

$f (1) = \sqrt{e^{1}}, - \sqrt{e^{1}}$

Step 12.2

Simplify the result.

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Step 12.2.1

Simplify.

$f (1) = \sqrt{e}, - \sqrt{e}$

Step 12.2.2

The final answer is

$\sqrt{e}, - \sqrt{e}$ .

$\sqrt{e}, - \sqrt{e}$

Step 12.3

Convert

$\sqrt{e}, - \sqrt{e}$ to decimal.

$y = \sqrt{e}, - \sqrt{e}$

Step 13

Find the point at

$x = 2$ .

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Step 13.1

Replace the variable

$x$ with

$2$ in the expression.

$f (2) = \sqrt{e^{2}}, - \sqrt{e^{2}}$

Step 13.2

Simplify the result.

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Step 13.2.1

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

$f (2) = e, - \sqrt{e^{2}}$

Step 13.2.2

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

$f (2) = e, - e$

Step 13.2.3

The final answer is

$e, - e$ .

$e, - e$

Step 13.3

Convert

$e, - e$ to decimal.

$y = e, - e$

Step 14

Find the point at

$x = 3$ .

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Step 14.1

Replace the variable

$x$ with

$3$ in the expression.

$f (3) = \sqrt{e^{3}}, - \sqrt{e^{3}}$

Step 14.2

Simplify the result.

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Step 14.2.1

Factor out

$e^{2}$ .

$f (3) = \sqrt{e^{2} e}, - \sqrt{e^{3}}$

Step 14.2.2

Pull terms out from under the radical.

$f (3) = e \sqrt{e}, - \sqrt{e^{3}}$

Step 14.2.3

Factor out

$e^{2}$ .

$f (3) = e \sqrt{e}, - \sqrt{e^{2} e}$

Step 14.2.4

Pull terms out from under the radical.

$f (3) = e \sqrt{e}, - e \sqrt{e}$

Step 14.2.5

The final answer is

$e \sqrt{e}, - e \sqrt{e}$ .

$e \sqrt{e}, - e \sqrt{e}$

Step 14.3

Convert

$e \sqrt{e}, - e \sqrt{e}$ to decimal.

$y = e \sqrt{e}, - e \sqrt{e}$

Step 15

Find the point at

$x = 4$ .

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Step 15.1

Replace the variable

$x$ with

$4$ in the expression.

$f (4) = \sqrt{e^{4}}, - \sqrt{e^{4}}$

Step 15.2

Simplify the result.

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Step 15.2.1

Rewrite

$e^{4}$ as

${(e^{2})}^{2}$ .

$f (4) = \sqrt{{(e^{2})}^{2}}, - \sqrt{e^{4}}$

Step 15.2.2

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

$f (4) = e^{2}, - \sqrt{e^{4}}$

Step 15.2.3

Rewrite

$e^{4}$ as

${(e^{2})}^{2}$ .

$f (4) = e^{2}, - \sqrt{{(e^{2})}^{2}}$

Step 15.2.4

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

$f (4) = e^{2}, - e^{2}$

Step 15.2.5

The final answer is

$e^{2}, - e^{2}$ .

$e^{2}, - e^{2}$

Step 15.3

Convert

$e^{2}, - e^{2}$ to decimal.

$y = e^{2}, - e^{2}$

Step 16

Find the point at

$x = 5$ .

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Step 16.1

Replace the variable

$x$ with

$5$ in the expression.

$f (5) = \sqrt{e^{5}}, - \sqrt{e^{5}}$

Step 16.2

Simplify the result.

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Step 16.2.1

Rewrite

$e^{5}$ as

${(e^{2})}^{2} e$ .

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Step 16.2.1.1

Factor out

$e^{4}$ .

$f (5) = \sqrt{e^{4} e}, - \sqrt{e^{5}}$

Step 16.2.1.2

Rewrite

$e^{4}$ as

${(e^{2})}^{2}$ .

$f (5) = \sqrt{{(e^{2})}^{2} e}, - \sqrt{e^{5}}$

Step 16.2.2

Pull terms out from under the radical.

$f (5) = e^{2} \sqrt{e}, - \sqrt{e^{5}}$

Step 16.2.3

Rewrite

$e^{5}$ as

${(e^{2})}^{2} e$ .

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Step 16.2.3.1

Factor out

$e^{4}$ .

$f (5) = e^{2} \sqrt{e}, - \sqrt{e^{4} e}$

Step 16.2.3.2

Rewrite

$e^{4}$ as

${(e^{2})}^{2}$ .

$f (5) = e^{2} \sqrt{e}, - \sqrt{{(e^{2})}^{2} e}$

Step 16.2.4

Pull terms out from under the radical.

$f (5) = e^{2} \sqrt{e}, - e^{2} \sqrt{e}$

Step 16.2.5

The final answer is

$e^{2} \sqrt{e}, - e^{2} \sqrt{e}$ .

$e^{2} \sqrt{e}, - e^{2} \sqrt{e}$

Step 16.3

Convert

$e^{2} \sqrt{e}, - e^{2} \sqrt{e}$ to decimal.

$y = e^{2} \sqrt{e}, - e^{2} \sqrt{e}$

Step 17

Find the point at

$x = 6$ .

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Step 17.1

Replace the variable

$x$ with

$6$ in the expression.

$f (6) = \sqrt{e^{6}}, - \sqrt{e^{6}}$

Step 17.2

Simplify the result.

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Step 17.2.1

Rewrite

$e^{6}$ as

${(e^{3})}^{2}$ .

$f (6) = \sqrt{{(e^{3})}^{2}}, - \sqrt{e^{6}}$

Step 17.2.2

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

$f (6) = e^{3}, - \sqrt{e^{6}}$

Step 17.2.3

Rewrite

$e^{6}$ as

${(e^{3})}^{2}$ .

$f (6) = e^{3}, - \sqrt{{(e^{3})}^{2}}$

Step 17.2.4

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

$f (6) = e^{3}, - e^{3}$

Step 17.2.5

The final answer is

$e^{3}, - e^{3}$ .

$e^{3}, - e^{3}$

Step 17.3

Convert

$e^{3}, - e^{3}$ to decimal.

$y = e^{3}, - e^{3}$

Step 18

Find the point at

$x = 7$ .

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Step 18.1

Replace the variable

$x$ with

$7$ in the expression.

$f (7) = \sqrt{e^{7}}, - \sqrt{e^{7}}$

Step 18.2

Simplify the result.

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Step 18.2.1

Rewrite

$e^{7}$ as

${(e^{3})}^{2} e$ .

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Step 18.2.1.1

Factor out

$e^{6}$ .

$f (7) = \sqrt{e^{6} e}, - \sqrt{e^{7}}$

Step 18.2.1.2

Rewrite

$e^{6}$ as

${(e^{3})}^{2}$ .

$f (7) = \sqrt{{(e^{3})}^{2} e}, - \sqrt{e^{7}}$

Step 18.2.2

Pull terms out from under the radical.

$f (7) = e^{3} \sqrt{e}, - \sqrt{e^{7}}$

Step 18.2.3

Rewrite

$e^{7}$ as

${(e^{3})}^{2} e$ .

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Step 18.2.3.1

Factor out

$e^{6}$ .

$f (7) = e^{3} \sqrt{e}, - \sqrt{e^{6} e}$

Step 18.2.3.2

Rewrite

$e^{6}$ as

${(e^{3})}^{2}$ .

$f (7) = e^{3} \sqrt{e}, - \sqrt{{(e^{3})}^{2} e}$

Step 18.2.4

Pull terms out from under the radical.

$f (7) = e^{3} \sqrt{e}, - e^{3} \sqrt{e}$

Step 18.2.5

The final answer is

$e^{3} \sqrt{e}, - e^{3} \sqrt{e}$ .

$e^{3} \sqrt{e}, - e^{3} \sqrt{e}$

Step 18.3

Convert

$e^{3} \sqrt{e}, - e^{3} \sqrt{e}$ to decimal.

$y = e^{3} \sqrt{e}, - e^{3} \sqrt{e}$

Step 19

Find the point at

$x = 8$ .

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Step 19.1

Replace the variable

$x$ with

$8$ in the expression.

$f (8) = \sqrt{e^{8}}, - \sqrt{e^{8}}$

Step 19.2

Simplify the result.

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Step 19.2.1

Rewrite

$e^{8}$ as

${(e^{4})}^{2}$ .

$f (8) = \sqrt{{(e^{4})}^{2}}, - \sqrt{e^{8}}$

Step 19.2.2

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

$f (8) = e^{4}, - \sqrt{e^{8}}$

Step 19.2.3

Rewrite

$e^{8}$ as

${(e^{4})}^{2}$ .

$f (8) = e^{4}, - \sqrt{{(e^{4})}^{2}}$

Step 19.2.4

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

$f (8) = e^{4}, - e^{4}$

Step 19.2.5

The final answer is

$e^{4}, - e^{4}$ .

$e^{4}, - e^{4}$

Step 19.3

Convert

$e^{4}, - e^{4}$ to decimal.

$y = e^{4}, - e^{4}$

Step 20

Find the point at

$x = 9$ .

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Step 20.1

Replace the variable

$x$ with

$9$ in the expression.

$f (9) = \sqrt{e^{9}}, - \sqrt{e^{9}}$

Step 20.2

Simplify the result.

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Step 20.2.1

Rewrite

$e^{9}$ as

${(e^{4})}^{2} e$ .

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Step 20.2.1.1

Factor out

$e^{8}$ .

$f (9) = \sqrt{e^{8} e}, - \sqrt{e^{9}}$

Step 20.2.1.2

Rewrite

$e^{8}$ as

${(e^{4})}^{2}$ .

$f (9) = \sqrt{{(e^{4})}^{2} e}, - \sqrt{e^{9}}$

Step 20.2.2

Pull terms out from under the radical.

$f (9) = e^{4} \sqrt{e}, - \sqrt{e^{9}}$

Step 20.2.3

Rewrite

$e^{9}$ as

${(e^{4})}^{2} e$ .

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Step 20.2.3.1

Factor out

$e^{8}$ .

$f (9) = e^{4} \sqrt{e}, - \sqrt{e^{8} e}$

Step 20.2.3.2

Rewrite

$e^{8}$ as

${(e^{4})}^{2}$ .

$f (9) = e^{4} \sqrt{e}, - \sqrt{{(e^{4})}^{2} e}$

Step 20.2.4

Pull terms out from under the radical.

$f (9) = e^{4} \sqrt{e}, - e^{4} \sqrt{e}$

Step 20.2.5

The final answer is

$e^{4} \sqrt{e}, - e^{4} \sqrt{e}$ .

$e^{4} \sqrt{e}, - e^{4} \sqrt{e}$

Step 20.3

Convert

$e^{4} \sqrt{e}, - e^{4} \sqrt{e}$ to decimal.

$y = e^{4} \sqrt{e}, - e^{4} \sqrt{e}$

Step 21

Find the point at

$x = 10$ .

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Step 21.1

Replace the variable

$x$ with

$10$ in the expression.

$f (10) = \sqrt{e^{10}}, - \sqrt{e^{10}}$

Step 21.2

Simplify the result.

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Step 21.2.1

Rewrite

$e^{10}$ as

${(e^{5})}^{2}$ .

$f (10) = \sqrt{{(e^{5})}^{2}}, - \sqrt{e^{10}}$

Step 21.2.2

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

$f (10) = e^{5}, - \sqrt{e^{10}}$

Step 21.2.3

Rewrite

$e^{10}$ as

${(e^{5})}^{2}$ .

$f (10) = e^{5}, - \sqrt{{(e^{5})}^{2}}$

Step 21.2.4

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

$f (10) = e^{5}, - e^{5}$

Step 21.2.5

The final answer is

$e^{5}, - e^{5}$ .

$e^{5}, - e^{5}$

Step 21.3

Convert

$e^{5}, - e^{5}$ to decimal.

$y = e^{5}, - e^{5}$

Step 22

The log function can be graphed using the vertical asymptote at

$y = 0$ and the points

$(1, 1.64872127), (1, - 1.64872127), (2, 2.71828182), (2, - 2.71828182), (3, 4.48168907), (3, - 4.48168907), (4, 7.38905609), (4, - 7.38905609), (5, 12.18249396), (5, - 12.18249396), (6, 20.08553692), (6, - 20.08553692), (7, 33.11545195), (7, - 33.11545195), (8, 54.59815003), (8, - 54.59815003), (9, 90.0171313), (9, - 90.0171313), (10, 148.4131591), (10, - 148.4131591), (1, \sqrt{e}, - \sqrt{e}), (2, e, - e), (3, e \sqrt{e}, - e \sqrt{e}), (4, e^{2}, - e^{2}), (5, e^{2} \sqrt{e}, - e^{2} \sqrt{e}), (6, e^{3}, - e^{3}), (7, e^{3} \sqrt{e}, - e^{3} \sqrt{e}), (8, e^{4}, - e^{4}), (9, e^{4} \sqrt{e}, - e^{4} \sqrt{e}), (10, e^{5}, - e^{5})$ .

Vertical Asymptote:

$y = 0$

Step 23