Calculus Examples

$y = \ln^{3} (x)$

Step 1

Interchange the variables.

$x = \ln^{3} (y)$

Step 2

Solve for $y$ .

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Step 2.1

Rewrite the equation as $\ln^{3} (y) = x$ .

$\ln^{3} (y) = x$

Step 2.2

Solve for $y$ .

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Step 2.2.1

Remove parentheses.

$\ln^{3} (y) = x$

Step 2.2.2

Solve for $y$ .

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Step 2.2.2.1

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

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

Step 2.2.2.2

To solve for $y$ , rewrite the equation using properties of logarithms.

$e^{\ln (y)} = e^{\sqrt[3]{x}}$

Step 2.2.2.3

Rewrite $\ln (y) = \sqrt[3]{x}$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$e^{\sqrt[3]{x}} = y$

Step 2.2.2.4

Rewrite the equation as $y = e^{\sqrt[3]{x}}$ .

$y = e^{\sqrt[3]{x}}$

Step 3

Replace $y$ with $f^{- 1} (x)$ to show the final answer.

$f^{- 1} (x) = e^{\sqrt[3]{x}}$

Step 4

Verify if $f^{- 1} (x) = e^{\sqrt[3]{x}}$ is the inverse of $f (x) = \ln^{3} (x)$ .

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Step 4.1

To verify the inverse, check if $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ .

Step 4.2

Evaluate $f^{- 1} (f (x))$ .

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Step 4.2.1

Set up the composite result function.

$f^{- 1} (f (x))$

Step 4.2.2

Evaluate $f^{- 1} (\ln^{3} (x))$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} (\ln^{3} (x)) = e^{\sqrt[3]{\ln^{3} (x)}}$

Step 4.2.3

Remove parentheses.

$f^{- 1} (\ln^{3} (x)) = e^{\sqrt[3]{\ln^{3} (x)}}$

Step 4.2.4

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

$f^{- 1} (\ln^{3} (x)) = e^{\ln (x)}$

Step 4.2.5

Exponentiation and log are inverse functions.

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

Step 4.3

Evaluate $f (f^{- 1} (x))$ .

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Step 4.3.1

Set up the composite result function.

$f (f^{- 1} (x))$

Step 4.3.2

Evaluate $f (e^{\sqrt[3]{x}})$ by substituting in the value of $f^{- 1}$ into $f$ .

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

Step 4.3.3

Use logarithm rules to move $\sqrt[3]{x}$ out of the exponent.

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

Step 4.3.4

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

$f (e^{\sqrt[3]{x}}) = {(\sqrt[3]{x} \cdot 1)}^{3}$

Step 4.3.5

Multiply $\sqrt[3]{x}$ by $1$ .

$f (e^{\sqrt[3]{x}}) = {\sqrt[3]{x}}^{3}$

Step 4.3.6

Rewrite ${\sqrt[3]{x}}^{3}$ as $x$ .

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Step 4.3.6.1

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

$f (e^{\sqrt[3]{x}}) = {(x^{\frac{1}{3}})}^{3}$

Step 4.3.6.2

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

$f (e^{\sqrt[3]{x}}) = x^{\frac{1}{3} \cdot 3}$

Step 4.3.6.3

Combine $\frac{1}{3}$ and $3$ .

$f (e^{\sqrt[3]{x}}) = x^{\frac{3}{3}}$

Step 4.3.6.4

Cancel the common factor of $3$ .

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Step 4.3.6.4.1

Cancel the common factor.

$f (e^{\sqrt[3]{x}}) = x^{\frac{3}{3}}$

Step 4.3.6.4.2

Rewrite the expression.

$f (e^{\sqrt[3]{x}}) = x$

Step 4.3.6.5

Simplify.

$f (e^{\sqrt[3]{x}}) = x$

Step 4.4

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = e^{\sqrt[3]{x}}$ is the inverse of $f (x) = \ln^{3} (x)$ .

$f^{- 1} (x) = e^{\sqrt[3]{x}}$