Find the Inverse y=e^(3x)

$y = e^{3 x}$

Step 1

Interchange the variables.

$x = e^{3 y}$

Step 2

Solve for $y$ .

Rewrite the equation as $e^{3 y} = x$ .

$e^{3 y} = x$

Take the natural logarithm of both sides of the equation to remove the variable from the exponent.

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

Expand the left side.

Expand $\ln (e^{3 y})$ by moving $3 y$ outside the logarithm.

$3 y \ln (e) = \ln (x)$

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

$3 y \cdot 1 = \ln (x)$

Multiply $3$ by $1$ .

$3 y = \ln (x)$

Divide each term in $3 y = \ln (x)$ by $3$ and simplify.

Divide each term in $3 y = \ln (x)$ by $3$ .

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

Simplify the left side.

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Step 3

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

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

Step 4

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

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

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

Set up the composite result function.

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

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

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

Expand $\ln (e^{3 x})$ by moving $3 x$ outside the logarithm.

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Divide $x \ln (e)$ by $1$ .

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

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

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

Multiply $x$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Cancel the common factor of $3$ .

Cancel the common factor.

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

Rewrite the expression.

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

Exponentiation and log are inverse functions.

$f (\frac{\ln (x)}{3}) = x$

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

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

Find the Inverse y=e^(3x)

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