Specifying the linear predictor, survival function and PDF of a log-normal survival distribution

Question

Continuing on my exploration of the log-normal distribution, I'm working on reimplementing some code originally written for a Weibull/Exponential model for a log-normal model. Among the things it does is use SAS's PROX NLMIXED to hard-calculate some likelihoods. While this is technically SAS code, I think it's pretty readable in a code-agnostic fashion.

For example, the Weibull Version:

*Linear Predictor of Event A
lam=exp(-(f0*alpha+f1*alpha*X));

*Density of Event A;
ff1=alpha*lam*t1**(alpha-1)*exp(-lam*t1**alpha);
ff2=alpha*lam*t2**(alpha-1)*exp(-lam*t2**alpha);

*Survival function of Event A;
sf1=exp(-lam*t1**alpha);
sf2=exp(-lam*t2**alpha);

The two survival functions are just at two different points in time for interval censoring.

I've implemented, I believe, the appropriate survival and density functions for a Log-Normal as follows:

*Density of EventA;
ff1 = exp(-0.5*((log(t1)-mu)/sigma)**2)/((t1*(2*CONSTANT('PI'))**0.5)*sigma);
ff2 = exp(-0.5*((log(t2)-mu)/sigma)**2)/((t2*(2*CONSTANT('PI'))**0.5)*sigma);

*Survival function of EventA;
sf1= 1 - CDF('Normal',((log(t1)-mu)/sigma));
sf2= 1 - CDF('Normal',((log(t2)-mu)/sigma));

I guess the first part of the question is does that appear correct? My second question is where I'm getting caught up, actually defining the linear predictor for the log normal. I think it should be something along the form of:

mu = exp(f0+f1*X) ala the Weibull above and with sigma estimated outside the linear predictor, but I'm not positive. Can someone point me in the right direction?

Edit: The goal PDF once the syntax errors are fixed is: