Let $X$ be a log-normal variate and $Y = aX + b$ is the affine transformation X.
Is $Y$ log-normal? I suspect it is not.
Since $X$ is log-normal, its expected value is
$$ E[X] = \exp(M + S^2/2) $$
The expected value of $Y$ is:
$$ E[Y] = aE[X] + b = a \exp(M + S^2/2) + b $$
To characterize $Y$ as a log-normal variate I should write its mean in the same form of $X$ mean, and obviously reproduce the same approach to others moments.
Does that make sense? or Is there another way to characterize $Y$ as log-normal? For example, using characteristic function.