inverse of f(x)=a^x — step-by-step solution

An exponential with a positive base other than 1 and a nonconstant affine exponent is strictly monotone on ℝ and takes every strictly positive value

Express input

x

in terms of output

y

y = a^{x}

⇥1) Let's solve in

ℝ

a^{x} = y

⇥The logarithm being defined only for positive numbers, and a base equal to 1 always giving 1, let's apply the ln function on both sides, provided that

a > 0

and

a \neq 1

and

y > 0

⇥

\ln (a^{x}) = \ln (y)

⇥

x \cdot \ln (a) = \ln (y)

⇥ⓘ Both sides are divided by

\ln (a)

, which must be nonzero:

a \neq 1

⇥Solution obtained:

x = \frac{\ln (y)}{\ln (a)}

⇥2) Solving for a:

a > 0

⇥Solution obtained:

a > 0

⇥3) Let's solve for a:

a \neq 1

using

a > 0

⇥Let's create the table of the different solutions to find the overall solution:

⇥

$a$	$-∞$		$0$		$1$		$+∞$
$a > 0$		❌	❌	✅	✅	✅
$a \neq 1$		✅	✅	✅	❌	✅
Global solution		❌	❌	✅	❌	✅

⇥Intersecting

a > 0

and

a \neq 1

gives

a \in (0, 1) \cup (1, +∞)

⇥4) Solving for y:

y > 0

⇥Solution found:

y > 0

Renaming the output as

x

, the inverse is

\frac{\ln (x)}{\ln (a)}

, with domain

{x > 0, a \in (0, 1) \cup (1, +∞)}

▶Inverse function:

\frac{\ln (x)}{\ln (a)}