inverse ln(-2x+3) — step-by-step solution

The function domain is

x < \frac{3}{2}

The logarithm of a nonconstant affine function is strictly monotone on its domain and takes every real value

Express input

x

in terms of output

y

y = \ln (- 2 x + 3)

⇥1) Let's solve in

ℝ

y = \ln (- 2 x + 3)

⇥

\ln (- 2 x + 3) = y

⇥Let's apply the exp function on both sides:

⇥

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

⇥ⓘ For real

x

\ln (\exp (x)) = x

; wherever

\ln (x)

is defined,

\exp (\ln (x)) = x

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

- 2 x + 3

⇥

- 2 x + 3 = e^{y}

⇥ⓘ We gather the terms in

x

on the right and the other terms on the left

⇥

3 - e^{y} = 2 x

⇥ⓘ Both sides are divided by

2

⇥Candidates, pending the domain check:

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

⇥2) Let's find the domain of

y = \ln (- 2 x + 3)

⇥ For the expression

\ln (- 2 x + 3)

to be defined, we must have:

- 2 x + 3 > 0

⇥ⓘ We add

2 x

to both sides

⇥

3 > 2 x

⇥ⓘ Both sides are divided by

2

⇥Domain constraints:

x < \frac{3}{2}

Renaming the output as

x

, the inverse is

\frac{3 - e^{x}}{2}

, with domain

x \in ℝ

▶Inverse function:

\frac{3 - e^{x}}{2}