Calculus Examples

$\ln (\frac{x^{4}}{{(3 x - 2)}^{3}})$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$\frac{x^{4}}{{(3 x - 2)}^{3}} = 0$

Step 1.2

Solve for

$x$ .

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Step 1.2.1

Set the numerator equal to zero.

$x^{4} = 0$

Step 1.2.2

Solve the equation for

$x$ .

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Step 1.2.2.1

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

$x = \pm \sqrt[4]{0}$

Step 1.2.2.2

Simplify

$\pm \sqrt[4]{0}$ .

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Step 1.2.2.2.1

Rewrite

$0$ as

$0^{4}$ .

$x = \pm \sqrt[4]{0^{4}}$

Step 1.2.2.2.2

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

$x = \pm 0$

Step 1.2.2.2.3

Plus or minus

$0$ is

$0$ .

$x = 0$

Step 1.3

The vertical asymptote occurs at

$x = 0$ .

Vertical Asymptote:

$x = 0$

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.

$f (1) = \ln (\frac{{(1)}^{4}}{{(3 (1) - 2)}^{3}})$

Step 2.2

Simplify the result.

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Step 2.2.1

One to any power is one.

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

Step 2.2.2

Simplify the denominator.

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Step 2.2.2.1

Multiply

$3$ by

$1$ .

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

Step 2.2.2.2

Subtract

$2$ from

$3$ .

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

Step 2.2.2.3

One to any power is one.

$f (1) = \ln (\frac{1}{1})$

Step 2.2.3

Divide

$1$ by

$1$ .

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

Step 2.2.4

The natural logarithm of

$1$ is

$0$ .

$f (1) = 0$

Step 2.2.5

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.

$f (2) = \ln (\frac{{(2)}^{4}}{{(3 (2) - 2)}^{3}})$

Step 3.2

Simplify the result.

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Step 3.2.1

Raise

$2$ to the power of

$4$ .

$f (2) = \ln (\frac{16}{{(3 (2) - 2)}^{3}})$

Step 3.2.2

Simplify the denominator.

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Step 3.2.2.1

Multiply

$3$ by

$2$ .

$f (2) = \ln (\frac{16}{{(6 - 2)}^{3}})$

Step 3.2.2.2

Subtract

$2$ from

$6$ .

$f (2) = \ln (\frac{16}{4^{3}})$

Step 3.2.2.3

Raise

$4$ to the power of

$3$ .

$f (2) = \ln (\frac{16}{64})$

Step 3.2.3

Cancel the common factor of

$16$ and

$64$ .

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Step 3.2.3.1

Factor

$16$ out of

$16$ .

$f (2) = \ln (\frac{16 (1)}{64})$

Step 3.2.3.2

Cancel the common factors.

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Step 3.2.3.2.1

Factor

$16$ out of

$64$ .

$f (2) = \ln (\frac{16 \cdot 1}{16 \cdot 4})$

Step 3.2.3.2.2

Cancel the common factor.

$f (2) = \ln (\frac{16 \cdot 1}{16 \cdot 4})$

Step 3.2.3.2.3

Rewrite the expression.

$f (2) = \ln (\frac{1}{4})$

Step 3.2.4

The final answer is

$\ln (\frac{1}{4})$ .

$\ln (\frac{1}{4})$

Step 3.3

Convert

$\ln (\frac{1}{4})$ to decimal.

$y = - 1.38629436$

Step 4

Find the point at

$x = 3$ .

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Step 4.1

Replace the variable

$x$ with

$3$ in the expression.

$f (3) = \ln (\frac{{(3)}^{4}}{{(3 (3) - 2)}^{3}})$

Step 4.2

Simplify the result.

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Step 4.2.1

Raise

$3$ to the power of

$4$ .

$f (3) = \ln (\frac{81}{{(3 (3) - 2)}^{3}})$

Step 4.2.2

Simplify the denominator.

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Step 4.2.2.1

Multiply

$3$ by

$3$ .

$f (3) = \ln (\frac{81}{{(9 - 2)}^{3}})$

Step 4.2.2.2

Subtract

$2$ from

$9$ .

$f (3) = \ln (\frac{81}{7^{3}})$

Step 4.2.2.3

Raise

$7$ to the power of

$3$ .

$f (3) = \ln (\frac{81}{343})$

Step 4.2.3

The final answer is

$\ln (\frac{81}{343})$ .

$\ln (\frac{81}{343})$

Step 4.3

Convert

$\ln (\frac{81}{343})$ to decimal.

$y = - 1.44328129$

Step 5

The log function can be graphed using the vertical asymptote at

$x = 0$ and the points

$(1, 0), (2, - 1.38629436), (3, - 1.44328129)$ .

Vertical Asymptote:

$x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & - 1.386 \\ 3 & - 1.443 \end{array}$

Step 6