Calculus Examples

$\log_{11} (x e^{x})$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x e^{x} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$e^{x} = 0$

Step 1.2.2

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.3

Set $e^{x}$ equal to $0$ and solve for $x$ .

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Step 1.2.3.1

Set $e^{x}$ equal to $0$ .

$e^{x} = 0$

Step 1.2.3.2

Solve $e^{x} = 0$ for $x$ .

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Step 1.2.3.2.1

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

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

Step 1.2.3.2.2

The equation cannot be solved because $\ln (0)$ is undefined.

Undefined

Step 1.2.3.2.3

There is no solution for $e^{x} = 0$

No solution

Step 1.2.4

The final solution is all the values that make $x e^{x} = 0$ true.

$x = 0$

Step 1.3

The vertical asymptote occurs at $x = 0$ .

Vertical Asymptote: $x = 0$

Step 2

Find the point at $x = 1$ .

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Step 2.1

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

$f (1) = \log_{11} ((1) \cdot e^{1})$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $e^{1}$ by $1$ .

$f (1) = \log_{11} (e)$

Step 2.2.2

The final answer is $\log_{11} (e)$ .

$\log_{11} (e)$

Step 2.3

Convert $\log_{11} (e)$ to decimal.

$y = 0.41703239$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = \log_{11} ((2) \cdot e^{2})$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $2$ by $e^{2}$ .

$f (2) = \log_{11} (2 e^{2})$

Step 3.2.2

The final answer is $\log_{11} (2 e^{2})$ .

$\log_{11} (2 e^{2})$

Step 3.3

Convert $\log_{11} (2 e^{2})$ to decimal.

$y = 1.1231296$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = \log_{11} ((3) \cdot e^{3})$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $3$ by $e^{3}$ .

$f (3) = \log_{11} (3 e^{3})$

Step 4.2.2

The final answer is $\log_{11} (3 e^{3})$ .

$\log_{11} (3 e^{3})$

Step 4.3

Convert $\log_{11} (3 e^{3})$ to decimal.

$y = 1.70925408$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0.41703239), (2, 1.1231296), (3, 1.70925408)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0.417 \\ 2 & 1.123 \\ 3 & 1.709 \end{array}$

Step 6