Log linear Bayesian hierarchical model specification in r using JAGS

Am familiar with model specification in this provided example with a Poisson distribution on N.lamb, N.ewe, and normal distribution on y.lamb, and y.ewe. I am interested in exploring this under a log-linear transformation. I am providing the sample if someone could correct the specification it will be greatly appreciated. Currently I have not altered the priors to a log scale but have altered in the likelihood portion of the code. Conversion (exponentiation) back to the original form will also be useful.

If someone could also incorporate a random explanatory variable for estimation of N.lamb, and N. ewe in the F and phi portion of the equation for illustrative purposes of transformations, that would also be greatly appreciated. Thanks.

library(jagsUI)

ewe_counts <- structure(list(Year = 1996:2015, Waffles = c(123L, 70L, 66L, 
NA, 51L, NA, 107L, 141L, NA, 118L, NA, 98L, NA, 103L, NA, 164L, 
141L, 90L, 96L, 60L)), class = "data.frame", row.names = c(NA, 
-20L))

lamb_counts <- structure(list(Year = 1996:2015, Waffles = c(70L, 31L, 19L, NA, 
27L, NA, 36L, 35L, NA, 26L, NA, 36L, NA, 34L, NA, 14L, 2L, 2L, 
21L, 20L)), class = "data.frame", row.names = c(NA, -20L))

model { 
  # Priors and constraints
    mu.lamb[1] ~ dunif(0, 300)
    sigma.lamb ~ dgamma(0.25, 0.25)  # SD 
    tau.lamb <- pow(sigma.lamb, -2) # precision
    N.lamb[1] ~ dnorm(mu.lamb[1],tau.lamb)  # Initial lamb abundance
    
    mu.ewe[1] ~ dunif(0, 300)
    sigma.ewe ~ dgamma(0.25, 0.25)  # SD 
    tau.ewe <- pow(sigma.ewe, -2) # precision
    N.ewe[1] ~ dnorm(mu.ewe[1],tau.ewe)   # Initial ewe abundance
    
    N.tot[1] <- N.lamb[1] + N.ewe[1] # Total abundance
    
  
  F ~ dgamma(0.25, 0.25) # Recruitment
  phi ~ dbeta(1, 1)      # Survival
  
  sigma.obs ~ dunif(0, 25)  # SD of observation process
  tau.obs <- pow(sigma.obs, -2)
  
  # Likelihood
    ###################
    ### State process
    ###################
    for (t in 2:(nYears)){
      N.tot[t] <- N.ewe[t] + N.lamb[t]
      
      ## Number of juveniles
      N.lamb[t] ~ dnorm(log(N.ewe[t-1] * F), tau.lamb)
      
      ## Number of adults
      N.ewe[t] ~ dnorm(log(N.tot[t-1] * phi), tau.ewe)
    } # end t loop
    
    ########################
    ### Observation process
    ########################
    for (t in 1:nYears) {
      y.lamb[t] ~ dnorm(log(N.lamb[t]), tau.obs)
      y.ewe[t] ~ dnorm(log(N.ewe[t]), tau.obs)
    }
}