inverse of y=ln(3x) — step-by-step solution

The function domain is

x > 0

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 (3 x)

⇥1) Let's solve in

ℝ

y = \ln (3 x)

⇥

\ln (3 x) = y

⇥You can apply the exp function on both sides:

⇥

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

⇥ⓘ For real

x

\ln (\exp (x)) = x

; wherever

\ln (x)

is defined,

\exp (\ln (x)) = x

e^{\ln (3 x)}

3 x

⇥

3 x = e^{y}

⇥ⓘ Both sides are divided by

3

⇥Candidates, pending the domain check:

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

⇥2) Let's find the domain of

y = \ln (3 x)

⇥ For the expression

\ln (3 x)

to exist, you must check:

3 x > 0

⇥We simplify the expression by

3

⇥⇥Let's solve in

ℝ

x > 0

⇥Domain constraints:

x > 0

Renaming the output as

x

, the inverse is

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

, with domain

x \in ℝ

▶Inverse function:

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