solve -e^x = y for x — step-by-step solution

Let's solve in

ℝ

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

ⓘ For real

x

\ln (\exp (x)) = x

; wherever

\ln (x)

is defined,

\exp (\ln (x)) = x

\ln (e^{x})

x

Solution obtained:

x = \ln (- y)

Solving for y:

- y > 0

ⓘ We add

y

to both sides

0 > y

Solution obtained:

y < 0

▶Solution:

{x = \ln (- y), y < 0}