Find the Inverse f(x)=3^x

$f (x) = 3^{x}$

Step 1

Write $f (x) = 3^{x}$ as an equation.

$y = 3^{x}$

Step 2

Interchange the variables.

$x = 3^{y}$

Step 3

Solve for $y$ .

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

$3^{y} = x$

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

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

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

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

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

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

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

Simplify the left side.

Cancel the common factor of $\ln (3)$ .

Cancel the common factor.

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

Divide $y$ by $1$ .

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

Step 4

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

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

Step 5

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

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 $f^{- 1} \cdot 3^{x}$ by substituting in the value of $f$ into $f^{- 1}$ .

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

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

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

Cancel the common factor of $\ln (3)$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

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

Set up the composite result function.

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

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

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

Use the change of base rule $\frac{\log_{b} (x)}{\log_{b} (a)} = \log_{a} (x)$ .

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

Exponentiation and log are inverse functions.

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

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

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

Find the Inverse f(x)=3^x

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