Find the Inverse natural log of 2x+3

$\ln (2 x) + 3$

Interchange the variables.

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

Solve for $y$ .

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Rewrite the equation as $\ln (2 y) + 3 = x$ .

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

Subtract $3$ from both sides of the equation.

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

To solve for $y$ , rewrite the equation using properties of logarithms.

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

Solve for $y$

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Exponentiation and log are inverse functions.

$2 y = e^{x - 3}$

Divide each term by $2$ and simplify.

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Divide each term in $2 y = e^{x - 3}$ by $2$ .

$\frac{2 y}{2} = \frac{e^{x - 3}}{2}$

Cancel the common factor of $2$ .

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Cancel the common factor.

$\frac{2 y}{2} = \frac{e^{x - 3}}{2}$

Divide $y$ by $1$ .

$y = \frac{e^{x - 3}}{2}$

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.

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

Set up the composite result function.

$g (f (x))$

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

$\frac{e^{(\ln (2 x) + 3) - 3}}{2}$

Simplify the numerator.

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Combine the opposite terms in $(\ln (2 x) + 3) - 3$ .

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Subtract $3$ from $3$ .

$g (\ln (2 x) + 3) = \frac{e^{\ln (2 x) + 0}}{2}$

Add $\ln (2 x)$ and $0$ .

$g (\ln (2 x) + 3) = \frac{e^{\ln (2 x)}}{2}$

Exponentiation and log are inverse functions.

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

Cancel the common factor of $2$ .

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Cancel the common factor.

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

Divide $x$ by $1$ .

$g (\ln (2 x) + 3) = x$

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

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

Find the Inverse natural log of 2x+3

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