inverse of f(x)=-2^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 = - 2^{x}

⇥1) Let's solve in

ℝ

- 2^{x} = y

⇥ⓘ We divide both sides by

-1

⇥The logarithm being defined only for positive numbers, let's apply the ln function on both sides, provided that

- y > 0

⇥

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

⇥ⓘ On the real domain of the original expression, the base is strictly positive: the logarithm of the power is the exponent times the logarithm of the base:

\ln (2^{x})

x \cdot \ln (2)

⇥

x \cdot \ln (2) = \ln (- y)

⇥ⓘ Both sides are divided by

\ln (2)

⇥Solution found:

x = \frac{\ln (- y)}{\ln (2)}

⇥2) Let's solve for y:

- y > 0

⇥ⓘ We add

y

to both sides

⇥

0 > y

⇥Solution found:

y < 0

Renaming the output as

x

, the inverse is

\frac{\ln (- x)}{\ln (2)}

, with domain

x < 0

, the condition found when isolating the input

▶Inverse function:

\frac{\ln (- x)}{\ln (2)}