How do/do not transforms on linear models preserve invertibility? In my generalized linear models class we did a natural logarithm transformation to a linear model in order to adjust its "quadratic characteristics" towards more linear ones.
This raised the question, after one's generated the model using logarithmic response (rather than the original response), can one reverse the logarithm in the resulting model to get back to the original scale or is this kind of invertibility destroyed in the model generating process?
I.e.
If one has a model $$Y \text{ ~ } X_1 + X_2 + \cdot \cdot \cdot$$
and one creates a model $$\hat{\log(Y)}$$ by taking $\log$ of the above response $Y$ and then creating the model, then can one use the resulting $\hat{\log(Y)}$ model to deduce back to $\hat{Y}$, i.e. deduce the model $\hat{Y}$ from model $\hat{\log(Y)}$ only (since in some sense $\hat{\log(Y)}$ has been gotten from $Y$ by taking $\log$, then can one go back by exponentiating?)?
 A: Not in general. Here's a quick example in R using the Anderson iris dataset:
data(iris)

# linear model
l1 <- lm(Petal.Length ~ Petal.Width, iris)
# log-linear model
l2 <- lm(log(Petal.Length) ~ Petal.Width, iris)

predict_length <- function(model, x) predict(model, data.frame(Petal.Width = x))

# linear prediction
pred1 <- function(x) predict_length(l1, x)

# exponentiated log-linear prediction
pred2 <- function(x) exp(predict_length(l2, x))

# lognormal "drop-in" estimator
# see: http://davegiles.blogspot.ca/2013/08/forecasting-from-log-linear-regressions.html
sigma2 <- summary(l2)$sigma
pred3 <- function(x) exp(predict_length(l2, x) + 0.5 * sigma2^2)

# compare fitted curves
plot(Petal.Length ~ Petal.Width, iris,
  col = 'darkgray', cex = 0.8,
  xlab = "Petal width", ylab = "Petal length",
  main = "Three fitted curves")
min_x <- min(iris$Petal.Width)
max_x <- max(iris$Petal.Width)
plot(pred1, min_x, max_x, add = TRUE, col = 1)
plot(pred2, min_x, max_x, add = TRUE, col = 2)
plot(pred3, min_x, max_x, add = TRUE, col = 3)
legend('bottomright', legend = c("Linear", "Loglinear naive", "Loglinear drop-in"), col = 1:3, lwd = 1)


