Find the Inverse f(x)=1.5^x+4

$f (x) = {1.5}^{x} + 4$

Replace $f (x)$ with $y$ .

$y = {1.5}^{x} + 4$

Interchange the variables.

$x = {1.5}^{y} + 4$

Solve for $y$ .

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Rewrite the equation as ${1.5}^{y} + 4 = x$ .

${1.5}^{y} + 4 = x$

Subtract $4$ from both sides of the equation.

${1.5}^{y} = x - 4$

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

$\ln ({1.5}^{y}) = \ln (x - 4)$

Expand $\ln ({1.5}^{y})$ by moving $y$ outside the logarithm.

$y \ln (1.5) = \ln (x - 4)$

Divide each term by $\ln (1.5)$ and simplify.

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Divide each term in $y \ln (1.5) = \ln (x - 4)$ by $\ln (1.5)$ .

$\frac{y \ln (1.5)}{\ln (1.5)} = \frac{\ln (x - 4)}{\ln (1.5)}$

Cancel the common factor of $\ln (1.5)$ .

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Cancel the common factor.

$\frac{y \ln (1.5)}{\ln (1.5)} = \frac{\ln (x - 4)}{\ln (1.5)}$

Divide $y$ by $1$ .

$y = \frac{\ln (x - 4)}{\ln (1.5)}$

Solve for $y$ and replace with $f^{- 1} (x)$ .

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

$f^{- 1} (x) = \frac{\ln (x - 4)}{\ln (1.5)}$

Set up the composite result function.

$g (f (x))$

Evaluate $g (f (x))$ by substituting in the value of $f$ into $g$ .

$g ({1.5}^{x} + 4) = \frac{\ln (({1.5}^{x} + 4) - 4)}{\ln (1.5)}$

Combine the opposite terms in $({1.5}^{x} + 4) - 4$ .

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Subtract $4$ from $4$ .

$g ({1.5}^{x} + 4) = \frac{\ln ({1.5}^{x} + 0)}{\ln (1.5)}$

Add ${1.5}^{x}$ and $0$ .

$g ({1.5}^{x} + 4) = \frac{\ln ({1.5}^{x})}{\ln (1.5)}$

Expand $\ln ({1.5}^{x})$ by moving $x$ outside the logarithm.

$g ({1.5}^{x} + 4) = \frac{x \ln (1.5)}{\ln (1.5)}$

Cancel the common factor of $\ln (1.5)$ .

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Cancel the common factor.

$g ({1.5}^{x} + 4) = \frac{x \ln (1.5)}{\ln (1.5)}$

Divide $x$ by $1$ .

$g ({1.5}^{x} + 4) = x$

Since $g (f (x)) = x$ , $f^{- 1} (x) = \frac{\ln (x - 4)}{\ln (1.5)}$ is the inverse of $f (x) = {1.5}^{x} + 4$ .

$f^{- 1} (x) = \frac{\ln (x - 4)}{\ln (1.5)}$

Find the Inverse f(x)=1.5^x+4

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