Find the Inverse y=(3/2)^x

$y = {(\frac{3}{2})}^{x}$

Interchange the variables.

$x = {(\frac{3}{2})}^{y}$

Solve for $y$ .

Rewrite the equation as ${(\frac{3}{2})}^{y} = x$ .

${(\frac{3}{2})}^{y} = x$

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

$\ln ({(\frac{3}{2})}^{y}) = \ln (x)$

Expand the left side.

Expand $\ln ({(\frac{3}{2})}^{y})$ by moving $y$ outside the logarithm.

$y \ln (\frac{3}{2}) = \ln (x)$

Rewrite $\ln (\frac{3}{2})$ as $\ln (3) - \ln (2)$ .

$y (\ln (3) - \ln (2)) = \ln (x)$

Simplify the left side.

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$y \ln (\frac{3}{2}) = \ln (x)$

Divide each term in $y \ln (\frac{3}{2}) = \ln (x)$ by $\ln (\frac{3}{2})$ and simplify.

Divide each term in $y \ln (\frac{3}{2}) = \ln (x)$ by $\ln (\frac{3}{2})$ .

$\frac{y \ln (\frac{3}{2})}{\ln (\frac{3}{2})} = \frac{\ln (x)}{\ln (\frac{3}{2})}$

Simplify the left side.

Cancel the common factor of $\ln (\frac{3}{2})$ .

Cancel the common factor.

$\frac{y \ln (\frac{3}{2})}{\ln (\frac{3}{2})} = \frac{\ln (x)}{\ln (\frac{3}{2})}$

Divide $y$ by $1$ .

$y = \frac{\ln (x)}{\ln (\frac{3}{2})}$

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

$f^{- 1} (x) = \frac{\ln (x)}{\ln (\frac{3}{2})}$

Verify if $f^{- 1} (x) = \frac{\ln (x)}{\ln (\frac{3}{2})}$ is the inverse of $f (x) = {(\frac{3}{2})}^{x}$ .

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 $f^{- 1} ({(\frac{3}{2})}^{x})$ by substituting in the value of $f$ into $f^{- 1}$ .

$f^{- 1} ({(\frac{3}{2})}^{x}) = \frac{\ln ({(\frac{3}{2})}^{x})}{\ln (\frac{3}{2})}$

Expand $\ln ({(\frac{3}{2})}^{x})$ by moving $x$ outside the logarithm.

$f^{- 1} ({(\frac{3}{2})}^{x}) = \frac{x \ln (\frac{3}{2})}{\ln (\frac{3}{2})}$

Cancel the common factor of $\ln (\frac{3}{2})$ .

Cancel the common factor.

$f^{- 1} ({(\frac{3}{2})}^{x}) = \frac{x \ln (\frac{3}{2})}{\ln (\frac{3}{2})}$

Divide $x$ by $1$ .

$f^{- 1} ({(\frac{3}{2})}^{x}) = x$

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

Set up the composite result function.

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

Evaluate $f (\frac{\ln (x)}{\ln (\frac{3}{2})})$ by substituting in the value of $f^{- 1}$ into $f$ .

$f (\frac{\ln (x)}{\ln (\frac{3}{2})}) = {(\frac{3}{2})}^{\frac{\ln (x)}{\ln (\frac{3}{2})}}$

Apply the product rule to $\frac{3}{2}$ .

$f (\frac{\ln (x)}{\ln (\frac{3}{2})}) = \frac{3^{\frac{\ln (x)}{\ln (\frac{3}{2})}}}{2^{\frac{\ln (x)}{\ln (\frac{3}{2})}}}$

Since $f^{- 1} (f (x)) = x$ and $f (f^{- 1} (x)) = x$ , then $f^{- 1} (x) = \frac{\ln (x)}{\ln (\frac{3}{2})}$ is the inverse of $f (x) = {(\frac{3}{2})}^{x}$ .

$f^{- 1} (x) = \frac{\ln (x)}{\ln (\frac{3}{2})}$

Find the Inverse y=(3/2)^x

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