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 $\log(Y+ΔY) = β_0 + β_1(X+ΔX)$?

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.
