Calculus Examples

$2 x \ln (x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x^{2} = 0$

Step 1.2

Solve for $x$ .

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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.

$x = \pm \sqrt{0}$

Step 1.2.2

Simplify $\pm \sqrt{0}$ .

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Step 1.2.2.1

Rewrite $0$ as $0^{2}$ .

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

Step 1.2.2.2

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

$x = \pm 0$

Step 1.2.2.3

Plus or minus $0$ is $0$ .

$x = 0$

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.

$f (1) = 2 (1) \ln (1)$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $2$ by $1$ .

$f (1) = 2 \ln (1)$

Step 2.2.2

The natural logarithm of $1$ is $0$ .

$f (1) = 2 \cdot 0$

Step 2.2.3

Multiply $2$ by $0$ .

$f (1) = 0$

Step 2.2.4

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

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

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $2$ by $2$ .

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

Step 3.2.2

Simplify $4 \ln (2)$ by moving $4$ inside the logarithm.

$f (2) = \ln (2^{4})$

Step 3.2.3

Raise $2$ to the power of $4$ .

$f (2) = \ln (16)$

Step 3.2.4

The final answer is $\ln (16)$ .

$\ln (16)$

Step 3.3

Convert $\ln (16)$ to decimal.

$y = 2.77258872$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = 2 (3) \ln (3)$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $2$ by $3$ .

$f (3) = 6 \ln (3)$

Step 4.2.2

Simplify $6 \ln (3)$ by moving $6$ inside the logarithm.

$f (3) = \ln (3^{6})$

Step 4.2.3

Raise $3$ to the power of $6$ .

$f (3) = \ln (729)$

Step 4.2.4

The final answer is $\ln (729)$ .

$\ln (729)$

Step 4.3

Convert $\ln (729)$ to decimal.

$y = 6.59167373$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (2, 2.77258872), (3, 6.59167373)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 2.773 \\ 3 & 6.592 \end{array}$

Step 6