Calculus Examples

$\ln ({(7 x^{9} + 4 x)}^{\frac{9}{5}})$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

${(7 x^{9} + 4 x)}^{\frac{9}{5}} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

Set the $7 x^{9} + 4 x$ equal to $0$ .

$7 x^{9} + 4 x = 0$

Step 1.2.2

Solve for $x$ .

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Step 1.2.2.1

Factor $x$ out of $7 x^{9} + 4 x$ .

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Step 1.2.2.1.1

Factor $x$ out of $7 x^{9}$ .

$x (7 x^{8}) + 4 x = 0$

Step 1.2.2.1.2

Factor $x$ out of $4 x$ .

$x (7 x^{8}) + x \cdot 4 = 0$

Step 1.2.2.1.3

Factor $x$ out of $x (7 x^{8}) + x \cdot 4$ .

$x (7 x^{8} + 4) = 0$

Step 1.2.2.2

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$7 x^{8} + 4 = 0$

Step 1.2.2.3

Set $x$ equal to $0$ .

$x = 0$

Step 1.2.2.4

Set $7 x^{8} + 4$ equal to $0$ and solve for $x$ .

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Step 1.2.2.4.1

Set $7 x^{8} + 4$ equal to $0$ .

$7 x^{8} + 4 = 0$

Step 1.2.2.4.2

Solve $7 x^{8} + 4 = 0$ for $x$ .

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Step 1.2.2.4.2.1

Subtract $4$ from both sides of the equation.

$7 x^{8} = - 4$

Step 1.2.2.4.2.2

Divide each term in $7 x^{8} = - 4$ by $7$ and simplify.

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Step 1.2.2.4.2.2.1

Divide each term in $7 x^{8} = - 4$ by $7$ .

$\frac{7 x^{8}}{7} = \frac{- 4}{7}$

Step 1.2.2.4.2.2.2

Simplify the left side.

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Step 1.2.2.4.2.2.2.1

Cancel the common factor of $7$ .

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Step 1.2.2.4.2.2.2.1.1

Cancel the common factor.

$\frac{7 x^{8}}{7} = \frac{- 4}{7}$

Step 1.2.2.4.2.2.2.1.2

Divide $x^{8}$ by $1$ .

$x^{8} = \frac{- 4}{7}$

Step 1.2.2.4.2.2.3

Simplify the right side.

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Step 1.2.2.4.2.2.3.1

Move the negative in front of the fraction.

$x^{8} = - \frac{4}{7}$

Step 1.2.2.4.2.3

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

$x = \pm \sqrt[8]{- \frac{4}{7}}$

Step 1.2.2.4.2.4

The complete solution is the result of both the positive and negative portions of the solution.

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Step 1.2.2.4.2.4.1

First, use the positive value of the $\pm$ to find the first solution.

$x = \sqrt[8]{- \frac{4}{7}}$

Step 1.2.2.4.2.4.2

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \sqrt[8]{- \frac{4}{7}}$

Step 1.2.2.4.2.4.3

The complete solution is the result of both the positive and negative portions of the solution.

$x = \sqrt[8]{- \frac{4}{7}}, - \sqrt[8]{- \frac{4}{7}}$

Step 1.2.2.5

The final solution is all the values that make $x (7 x^{8} + 4) = 0$ true.

$x = 0, \sqrt[8]{- \frac{4}{7}}, - \sqrt[8]{- \frac{4}{7}}$

Step 1.2.3

Exclude the solutions that do not make ${(7 x^{9} + 4 x)}^{\frac{9}{5}} = 0$ true.

$x = 0$

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.

$f (1) = \ln ({(7 {(1)}^{9} + 4 (1))}^{\frac{9}{5}})$

Step 2.2

Simplify the result.

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Step 2.2.1

Simplify each term.

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Step 2.2.1.1

One to any power is one.

$f (1) = \ln ({(7 \cdot 1 + 4 (1))}^{\frac{9}{5}})$

Step 2.2.1.2

Multiply $7$ by $1$ .

$f (1) = \ln ({(7 + 4 (1))}^{\frac{9}{5}})$

Step 2.2.1.3

Multiply $4$ by $1$ .

$f (1) = \ln ({(7 + 4)}^{\frac{9}{5}})$

Step 2.2.2

Add $7$ and $4$ .

$f (1) = \ln (11^{\frac{9}{5}})$

Step 2.2.3

The final answer is $\ln (11^{\frac{9}{5}})$ .

$\ln (11^{\frac{9}{5}})$

Step 2.3

Convert $\ln (11^{\frac{9}{5}})$ to decimal.

$y = 4.31621149$

Step 3

Find the point at $x = 2$ .

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Step 3.1

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

$f (2) = \ln ({(7 {(2)}^{9} + 4 (2))}^{\frac{9}{5}})$

Step 3.2

Simplify the result.

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Step 3.2.1

Simplify each term.

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Step 3.2.1.1

Raise $2$ to the power of $9$ .

$f (2) = \ln ({(7 \cdot 512 + 4 (2))}^{\frac{9}{5}})$

Step 3.2.1.2

Multiply $7$ by $512$ .

$f (2) = \ln ({(3584 + 4 (2))}^{\frac{9}{5}})$

Step 3.2.1.3

Multiply $4$ by $2$ .

$f (2) = \ln ({(3584 + 8)}^{\frac{9}{5}})$

Step 3.2.2

Add $3584$ and $8$ .

$f (2) = \ln (3592^{\frac{9}{5}})$

Step 3.2.3

The final answer is $\ln (3592^{\frac{9}{5}})$ .

$\ln (3592^{\frac{9}{5}})$

Step 3.3

Convert $\ln (3592^{\frac{9}{5}})$ to decimal.

$y = 14.73563597$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = \ln ({(7 {(3)}^{9} + 4 (3))}^{\frac{9}{5}})$

Step 4.2

Simplify the result.

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Step 4.2.1

Simplify each term.

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Step 4.2.1.1

Raise $3$ to the power of $9$ .

$f (3) = \ln ({(7 \cdot 19683 + 4 (3))}^{\frac{9}{5}})$

Step 4.2.1.2

Multiply $7$ by $19683$ .

$f (3) = \ln ({(137781 + 4 (3))}^{\frac{9}{5}})$

Step 4.2.1.3

Multiply $4$ by $3$ .

$f (3) = \ln ({(137781 + 12)}^{\frac{9}{5}})$

Step 4.2.2

Add $137781$ and $12$ .

$f (3) = \ln (137793^{\frac{9}{5}})$

Step 4.2.3

The final answer is $\ln (137793^{\frac{9}{5}})$ .

$\ln (137793^{\frac{9}{5}})$

Step 4.3

Convert $\ln (137793^{\frac{9}{5}})$ to decimal.

$y = 21.3003141$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 4.31621149), (2, 14.73563597), (3, 21.3003141)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 4.316 \\ 2 & 14.736 \\ 3 & 21.3 \end{array}$

Step 6