Calculus Examples

$9 x \ln (8 x)$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$134217728 x^{9} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Divide each term in $134217728 x^{9} = 0$ by $134217728$ and simplify.

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Step 1.2.1.1

Divide each term in $134217728 x^{9} = 0$ by $134217728$ .

$\frac{134217728 x^{9}}{134217728} = \frac{0}{134217728}$

Step 1.2.1.2

Simplify the left side.

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Step 1.2.1.2.1

Cancel the common factor of $134217728$ .

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Step 1.2.1.2.1.1

Cancel the common factor.

$\frac{134217728 x^{9}}{134217728} = \frac{0}{134217728}$

Step 1.2.1.2.1.2

Divide $x^{9}$ by $1$ .

$x^{9} = \frac{0}{134217728}$

Step 1.2.1.3

Simplify the right side.

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Step 1.2.1.3.1

Divide $0$ by $134217728$ .

$x^{9} = 0$

Step 1.2.2

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

$x = \sqrt[9]{0}$

Step 1.2.3

Simplify $\sqrt[9]{0}$ .

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Step 1.2.3.1

Rewrite $0$ as $0^{9}$ .

$x = \sqrt[9]{0^{9}}$

Step 1.2.3.2

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

$x = 0$

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.

$f (1) = 9 (1) \ln (8 (1))$

Step 2.2

Simplify the result.

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Step 2.2.1

Multiply $9$ by $1$ .

$f (1) = 9 \ln (8 (1))$

Step 2.2.2

Multiply $8$ by $1$ .

$f (1) = 9 \ln (8)$

Step 2.2.3

Simplify $9 \ln (8)$ by moving $9$ inside the logarithm.

$f (1) = \ln (8^{9})$

Step 2.2.4

Raise $8$ to the power of $9$ .

$f (1) = \ln (134217728)$

Step 2.2.5

The final answer is $\ln (134217728)$ .

$\ln (134217728)$

Step 2.3

Convert $\ln (134217728)$ to decimal.

$y = 18.71497387$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = 9 (2) \ln (8 (2))$

Step 3.2

Simplify the result.

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Step 3.2.1

Multiply $9$ by $2$ .

$f (2) = 18 \ln (8 (2))$

Step 3.2.2

Multiply $8$ by $2$ .

$f (2) = 18 \ln (16)$

Step 3.2.3

Simplify $18 \ln (16)$ by moving $18$ inside the logarithm.

$f (2) = \ln (16^{18})$

Step 3.2.4

Raise $16$ to the power of $18$ .

$f (2) = \ln (4722366482869640000000)$

Step 3.2.5

The final answer is $\ln (4722366482869640000000)$ .

$\ln (4722366482869640000000)$

Step 3.3

Convert $\ln (4722366482869640000000)$ to decimal.

$y = 49.906597$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = 9 (3) \ln (8 (3))$

Step 4.2

Simplify the result.

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Step 4.2.1

Multiply $9$ by $3$ .

$f (3) = 27 \ln (8 (3))$

Step 4.2.2

Multiply $8$ by $3$ .

$f (3) = 27 \ln (24)$

Step 4.2.3

Simplify $27 \ln (24)$ by moving $27$ inside the logarithm.

$f (3) = \ln (24^{27})$

Step 4.2.4

The final answer is $\ln (24^{27})$ .

$\ln (24^{27})$

Step 4.3

Convert $\ln (24^{27})$ to decimal.

$y = 85.80745341$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 18.71497387), (2, 49.906597), (3, 85.80745341)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 18.715 \\ 2 & 49.907 \\ 3 & 85.807 \end{array}$

Step 6