Is ln(y+Δy) == β0 + β1(X+ΔX) in log-linear/non-linear regression? I came across the following in explaining the log-linear regression model.
Given the model $\log(Y_i) = β_0 + β_1X_i + u_i$
The expected value of $\log(Y)$ given $X$ is $β_0 + β_1X$.
So far, so good. But then it says:

'When $X$ is $X+ΔX$, the expected value is given by $\log(Y+ΔY)$'.

I don't see why this is necessarily the case. Could someone explain why $\log(Y+ΔY) = β_0 + β_1(X+ΔX)$?
 A: Your question is complete without defining what $\Delta Y$ is. Regardless, a tick thing about log-linear regression is that if you want to get the expected value after anti-log transformation, a bias has to be corrected. Details are given in many classical papers (e.g., Beauchamp, J.J. and Olson, J.S., 1973. Corrections for bias in regression estimates after logarithmic transformation. Ecology, 54(6), pp.1403-1407; Sprugel, D. G. "Correctiong for bias in log-transformed allometric equations." Ecology 64 (1983): 209-210; Newman MC. Regression analysis of log‐transformed data: Statistical bias and its correction. Environmental Toxicology and Chemistry: An International Journal. 1993 Jun;12(6):1129-33.)
Here is the gist: suppose that $ Z=log(Y)=\beta_0+\beta_1*X+u$. The expected value of Z or log )Y) is simply  $E[Z]=E[log(Y)]=\beta_0+\beta_1*X$, but the expectation of Y is not $exp(\beta_0+\beta_1*X)$, but a more correct one is $E(Y)=exp(\beta_0+\beta_1*X+{\sigma}^2/2)$.   Whatever you are trying to derive, if it involves anti-log transformation and taking expectation, this bias correction should not be ignored.
A: IF the model is $\log(Y_i) = β_0 + β_1X_i + u_i$ then it follows that for another $X_j$
$$\log(Y_j) = β_0 + β_1X_j + u_j$$
We can define then $Y_j= Y_i + \Delta Y $, with $ (Y_j > 0, Y_i > 0)$, and $X_j=X_i + \Delta X$.
This implies that
$$\log(Y_i + \Delta Y) = β_0 + β_1(X_i + \Delta X) + u_j$$
Notice, however, the error term in this form: it's $u_j$, not $u_i$.
You can make arguments regarding the expected value from here, given $X_i, \Delta X$ and model assumptions.

As @whuber rightfully pointed in the comments, this is a general result that is valid under linearity (of the independent variables).
You can see this because the solution is completely agnostic to the fact that the link function is a logarithm. It would've worked the same under any function.
