Algebra Examples

f (x) = ln (x)

$f (x) = \ln (x)$

Step 1

Write

f (x) = ln (x)

$f (x) = \ln (x)$ as an equation.

y = ln (x)

$y = \ln (x)$

Step 2

Interchange the variables.

x = ln (y)

$x = \ln (y)$

Step 3

Solve for

y

$y$ .

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Step 3.1

Rewrite the equation as

ln (y) = x

$\ln (y) = x$ .

ln (y) = x

$\ln (y) = x$

Step 3.2

To solve for

y

$y$ , rewrite the equation using properties of logarithms.

e^{ln (y)} = e^{x}

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

Step 3.3

Rewrite

ln (y) = x

$\ln (y) = x$ in exponential form using the definition of a logarithm. If

x

$x$ and

b

$b$ are positive real numbers and

b \neq 1

$b \neq 1$ , then

{log}_{b} (x) = y

$\log_{b} (x) = y$ is equivalent to

b^{y} = x

$b^{y} = x$ .

e^{x} = y

$e^{x} = y$

Step 3.4

Rewrite the equation as

y = e^{x}

$y = e^{x}$ .

y = e^{x}

$y = e^{x}$

y = e^{x}

$y = e^{x}$

Step 4

Replace

y

$y$ with

f^{- 1} (x)

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

f^{- 1} (x) = e^{x}

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

Step 5

Verify if

f^{- 1} (x) = e^{x}

$f^{- 1} (x) = e^{x}$ is the inverse of

f (x) = ln (x)

$f (x) = \ln (x)$ .

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Step 5.1

To verify the inverse, check if

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

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

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

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

Step 5.2

Evaluate

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

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

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Step 5.2.1

Set up the composite result function.

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

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

Step 5.2.2

Evaluate

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

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

f

$f$ into

f^{- 1}

$f^{- 1}$ .

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

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

Step 5.2.3

Exponentiation and log are inverse functions.

f^{- 1} (ln (x)) = x

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

f^{- 1} (ln (x)) = x

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

Step 5.3

Evaluate

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

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

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Step 5.3.1

Set up the composite result function.

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

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

Step 5.3.2

Evaluate

f (e^{x})

$f (e^{x})$ by substituting in the value of

f^{- 1}

$f^{- 1}$ into

f

$f$ .

f (e^{x}) = ln (e^{x})

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

Step 5.3.3

Use logarithm rules to move

x

$x$ out of the exponent.

f (e^{x}) = x ln (e)

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

Step 5.3.4

The natural logarithm of

e

$e$ is

1

$1$ .

f (e^{x}) = x \cdot 1

$f (e^{x}) = x \cdot 1$

Step 5.3.5

Multiply

x

$x$ by

1

$1$ .

f (e^{x}) = x

$f (e^{x}) = x$

f (e^{x}) = x

$f (e^{x}) = x$

Step 5.4

Since

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

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

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

$f (f^{- 1} (x)) = x$ , then

f^{- 1} (x) = e^{x}

$f^{- 1} (x) = e^{x}$ is the inverse of

f (x) = ln (x)

$f (x) = \ln (x)$ .

f^{- 1} (x) = e^{x}

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

f^{- 1} (x) = e^{x}

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