Calculus Examples

$\ln (\sqrt[3]{x})$

Step 1

Find the asymptotes.

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Step 1.1

Set the argument of the logarithm equal to zero.

$\sqrt[3]{x} = 0$

Step 1.2

Solve for $x$ .

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Step 1.2.1

To remove the radical on the left side of the equation, cube both sides of the equation.

${\sqrt[3]{x}}^{3} = 0^{3}$

Step 1.2.2

Simplify each side of the equation.

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Step 1.2.2.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt[3]{x}$ as $x^{\frac{1}{3}}$ .

${(x^{\frac{1}{3}})}^{3} = 0^{3}$

Step 1.2.2.2

Simplify the left side.

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Step 1.2.2.2.1

Simplify ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.2.2.1.1

Multiply the exponents in ${(x^{\frac{1}{3}})}^{3}$ .

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Step 1.2.2.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.2.2.2.1.1.2

Cancel the common factor of $3$ .

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Step 1.2.2.2.1.1.2.1

Cancel the common factor.

$x^{\frac{1}{3} \cdot 3} = 0^{3}$

Step 1.2.2.2.1.1.2.2

Rewrite the expression.

$x^{1} = 0^{3}$

Step 1.2.2.2.1.2

Simplify.

$x = 0^{3}$

Step 1.2.2.3

Simplify the right side.

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Step 1.2.2.3.1

Raising $0$ to any positive power yields $0$ .

$x = 0$

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.

$f (1) = \ln (\sqrt[3]{1})$

Step 2.2

Simplify the result.

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Step 2.2.1

Remove parentheses.

$f (1) = \ln (\sqrt[3]{1})$

Step 2.2.2

Any root of $1$ is $1$ .

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

Step 2.2.3

The natural logarithm of $1$ is $0$ .

$f (1) = 0$

Step 2.2.4

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.

$f (2) = \ln (\sqrt[3]{2})$

Step 3.2

Simplify the result.

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Step 3.2.1

Remove parentheses.

$f (2) = \ln (\sqrt[3]{2})$

Step 3.2.2

The final answer is $\ln (\sqrt[3]{2})$ .

$\ln (\sqrt[3]{2})$

Step 3.3

Convert $\ln (\sqrt[3]{2})$ to decimal.

$y = 0.23104906$

Step 4

Find the point at $x = 3$ .

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Step 4.1

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

$f (3) = \ln (\sqrt[3]{3})$

Step 4.2

Simplify the result.

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Step 4.2.1

Remove parentheses.

$f (3) = \ln (\sqrt[3]{3})$

Step 4.2.2

The final answer is $\ln (\sqrt[3]{3})$ .

$\ln (\sqrt[3]{3})$

Step 4.3

Convert $\ln (\sqrt[3]{3})$ to decimal.

$y = 0.36620409$

Step 5

The log function can be graphed using the vertical asymptote at $x = 0$ and the points $(1, 0), (2, 0.23104906), (3, 0.36620409)$ .

Vertical Asymptote: $x = 0$

$\begin{array}{c} x & y \\ 1 & 0 \\ 2 & 0.231 \\ 3 & 0.366 \end{array}$

Step 6