It it possible in R to specify a regression formula for the hazard rate for a survival analysis model? I am currently trying to fit a survival analysis model which has the following survival function:
$S(t) = \lambda_i e^{-\lambda_i t}$
but with 
$\lambda_i = e^{\beta_0 +\beta_1 log(1+X_i)}$
where $X_i$ represents each unique observation. 
I am trying to use the default survival package but am not sure where I can program in the regression formula for the hazard rate. I believe I might be misunderstanding what the survival analysis package does but would anyone be able to tell me if this can be even done in the R package? Thanks!
 A: You seem to be talking about a simple parametric survival model.
An exponential survival model would have survivor function $S(t)=e^{-\lambda_it}$, and hazard rate $\lambda_i$. (The function $f_i(t)= \lambda_ie^{-\lambda_it}$ would be the failure density.)
A parametric survival model is akin to a regression or a GLM, in that it has linear predictors, but (besides the fact that you're modelling survival time) the big difference is it's set up to deal with censoring. If there's no censoring an exponential model fitted in a GLM package should give essentially the same answers as one fitted using a survival package.
If you just use "$\log(1+x_i)$" as your time, $t$, in the survival model, what you'll get is a model where the survival probability decreases with $x$ (with constant hazard rate per unit change in $\log(1+x)$)
This could be done in R via the survreg function.
(However, it might be better to explain what you're trying to achieve; it's a little hard to judge if what you're doing makes sense, because there's not much detail.)
After loading the survival package, see ?survreg for information on how to call the function. You will want to set the dist argument to fit the particular $S(t)$ you mentioned.
There's an example of using the particular 'dist' function you need in the help which uses the built in data set "ovarian".
You'll note (from reading the examples) that to fit a survival model you call the Surv function inside the survreg call. This is to set up the censoring information to go with the survival times.
You'll also need to pass the correct form of the predictor, $X$ to get the model you mention in your question.
I'd suggest trying the examples in the help (while you can cut and paste, there's probably more benefit in actually typing them in one by one).
There are numerous worked examples of using survreg (both with this survivor function and with others) on the internet. 
There are examples here, here, here that you might like to take a look at (and you can probably find better ones if you look around).
This is not really the place to get an R tutorial though. Specific questions about survival analysis are likely to fare better (possibly involving R as long as it's not essentially a purely coding question), but you may be better looking for some tutorials on basic survival analysis to begin, in order to be able to ask specific questions. 
