Precalculus Examples

$f (x) = x \log (x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$x = 0$

Step 1.2

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.

$f (1) = (1) \log (1)$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $\log (1)$ by $1$ .

$f (1) = \log (1)$

Step 2.2.2

Logarithm base $10$ of $1$ is $0$ .

$f (1) = 0$

Step 2.2.3

The final answer is $0$ .

$0$

Step 2.3

Convert $0$ to decimal.

$y = 0$

Step 3

Find the point at $x = 10$ .

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Step 3.1

Replace the variable $x$ with $10$ in the expression.

$f (10) = (10) \log (10)$

Step 3.2

Simplify the result.

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Step 3.2.1

Logarithm base $10$ of $10$ is $1$ .

$f (10) = 10 \cdot 1$

Step 3.2.2

Multiply $10$ by $1$ .

$f (10) = 10$

Step 3.2.3

The final answer is $10$ .

$10$

Step 3.3

Convert $10$ to decimal.

$y = 10$

Step 4

Find the point at $x = 2$ .

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Step 4.1

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

$f (2) = (2) \log (2)$

Step 4.2

Simplify the result.

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Step 4.2.1

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

$f (2) = \log (2^{2})$

Step 4.2.2

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

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

Step 4.2.3

The final answer is $\log (4)$ .

$\log (4)$

Step 4.3

Convert $\log (4)$ to decimal.

$y = 0.60205999$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (10, 10), (2, 0.60205999)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.602 \\ 10 & 10 \end{array}$

Step 6