What is the PDF, CDF, and E[Y] of Y=ln[X+c] if X is lognormal

Question

If $\ln X \sim N(\mu, \sigma^2)$, what is the distribution of $Y=\ln \left(X+c\right)$ where $c$ is a constant. Is this something that can be written out analytically?

Also, what is $E[Y]$?

Answer 1

Hint: $$P(\ln(X + c) \le x) = P(X \le e^x - c) = \begin{cases} 0 & e^x \le c \\ P(\ln(X) \le \ln(e^x - c)) & \text{else} \end{cases}$$