Cox Regression Model for toxicity

Question

I'm trying to evaluate the toxicity of a certain treatment over time. I have 2 independent groups and I apply a treatment to one, and nothing to the second one. I thought about using logistic regression but since in the control group all of them are alive at the end, I would have a problem of separation, so I was trying to look for an alternative.

I've 6 time points (0h,2h,4h,6h,8h and 24h) and the data is binary (dead=0, alive=1). I was wondering if I could use the COX regression model to test if this treatment is toxic for this population over time, using R:

res.cox <- coxph(Surv( time, alive_dead) ~treatment, data=data_aliveD) summary(res.cox)

I get this:

Call:
coxph(formula = Surv(time, alive_dead) ~ treatment, data = data_aliveD)

  n= 720, number of events= 654 

                    coef exp(coef) se(coef)      z Pr(>|z|)    
treatment T -0.30952   0.73380  0.07952 -3.893 9.92e-05 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

                exp(coef) exp(-coef) lower .95 upper .95
treatment  T    0.7338      1.363    0.6279    0.8576

Concordance= 0.514  (se = 0.014 )
Rsquare= 0.021   (max possible= 1 )
Likelihood ratio test= 15.27  on 1 df,   p=9e-05
Wald test            = 15.15  on 1 df,   p=1e-04
Score (logrank) test = 15.27  on 1 df,   p=9e-05

Also I would like to know what is the difference between Cox Regression Model and Log-Rank test. If I use Long-rank test I have the following output:

log.rank <- survdiff(Surv( time, alive_dead) ~treatment, data=data_aliveD)

log.rank

Call:
survdiff(formula = Surv(time, alive_dead) ~ treatment, data = data_aliveD)

                   N Observed Expected (O-E)^2/E (O-E)^2/V
treatment=H2O        360      360      327      3.33       9.7
treatment=T 360      294      327      3.33       9.7

 Chisq= 9.7  on 1 degrees of freedom, p= 0.002 

Thank you!