Pre-Algebra Examples

$x^{2} \ln (3 x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$3 x = 0$

Step 1.2

Divide each term in $3 x = 0$ by $3$ and simplify.

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Step 1.2.1

Divide each term in $3 x = 0$ by $3$ .

$\frac{3 x}{3} = \frac{0}{3}$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $3$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{0}{3}$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{0}{3}$

Step 1.2.3

Simplify the right side.

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Step 1.2.3.1

Divide $0$ by $3$ .

$x = 0$

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.

$f (1) = {(1)}^{2} \ln (3 (1))$

Step 2.2

Simplify the result.

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Step 2.2.1

One to any power is one.

$f (1) = 1 \ln (3 (1))$

Step 2.2.2

Multiply $\ln (3 (1))$ by $1$ .

$f (1) = \ln (3 (1))$

Step 2.2.3

Multiply $3$ by $1$ .

$f (1) = \ln (3)$

Step 2.2.4

The final answer is $\ln (3)$ .

$\ln (3)$

Step 2.3

Convert $\ln (3)$ to decimal.

$y = 1.09861228$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = {(2)}^{2} \ln (3 (2))$

Step 3.2

Simplify the result.

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Step 3.2.1

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

$f (2) = 4 \ln (3 (2))$

Step 3.2.2

Multiply $3$ by $2$ .

$f (2) = 4 \ln (6)$

Step 3.2.3

Simplify $4 \ln (6)$ by moving $4$ inside the logarithm.

$f (2) = \ln (6^{4})$

Step 3.2.4

Raise $6$ to the power of $4$ .

$f (2) = \ln (1296)$

Step 3.2.5

The final answer is $\ln (1296)$ .

$\ln (1296)$

Step 3.3

Convert $\ln (1296)$ to decimal.

$y = 7.16703787$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = {(3)}^{2} \ln (3 (3))$

Step 4.2

Simplify the result.

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Step 4.2.1

Raise $3$ to the power of $2$ .

$f (3) = 9 \ln (3 (3))$

Step 4.2.2

Multiply $3$ by $3$ .

$f (3) = 9 \ln (9)$

Step 4.2.3

Simplify $9 \ln (9)$ by moving $9$ inside the logarithm.

$f (3) = \ln (9^{9})$

Step 4.2.4

Raise $9$ to the power of $9$ .

$f (3) = \ln (387420489)$

Step 4.2.5

The final answer is $\ln (387420489)$ .

$\ln (387420489)$

Step 4.3

Convert $\ln (387420489)$ to decimal.

$y = 19.77502119$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 1.09861228), (2, 7.16703787), (3, 19.77502119)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 1.099 \\ 2 & 7.167 \\ 3 & 19.775 \end{array}$

Step 6