I'm trying to find the inverse transform of the exponential CDF to sample from it (make x the subject of the equation)
see the examples section of https://en.wikipedia.org/wiki/Inverse_transform_sampling
Attempt 1:
$$ y=1-e^{-\lambda x} $$
$$ 1-y = e^{-\lambda x} $$
$$ log(1-y) = -\lambda x $$
$$ x = -log(1-y)/\lambda $$
Attempt 2:
$$ y=1-e^{-\lambda x} $$
$$ y-1 = -e^{-\lambda x} $$
$$ log(y-1) = \lambda x $$
$$ x = log(y-1)/\lambda $$
question: Is $ -log(1-y)/\lambda \equiv log(y-1)/\lambda $
If not, what have I done wrong?