Log-log linear model interpretation, when does the approximation not hold? I read in the following post: Interpreting Log-Transformed Percentages in OLS that

In a log-log model, such as $\log(y) = b_0 + b_1 \log(x)$, I know that
with OLS the standard interpretation is a "1% increase in x is
associated with a $b_1$% increase in y."

I have seen this statement everywhere and it is the standard interpretation, as the poster says. However, from what I understand the interpretation is just an approximation:
$$
\ln(Y+\Delta Y)-\ln(Y) = \beta_1\left[(\ln(X+\Delta X)-\ln(X)\right]
$$
Then, the approximation occurs when we have
$$
\frac{\Delta Y}{Y} \approx\beta_1 \frac{\Delta X}{X}
$$
(the approximation is from calculus when $\frac{\Delta x}{x}$ is small)
Solving for $\beta_1$ we get:
$$
\beta_1 = \dfrac{\Delta Y / Y}{\Delta X / X}
$$
which is the percentage interpretation we commonly see. In most settings I have seen, no one talks about this as an approximation, but rather speak of it as if it holds in equality. Is such an approximation usually robust in applied settings?

Using differentials
$$\beta_1 = \frac{d\log y}{d\log x} = \frac{(1/y)dy}{(1/x)dx} = \frac{dy/y}{dx/x} \approx \frac{\Delta y/y}{\Delta x/x}$$.
So the approximation is as accurate as a point expression being used for intervals. "Small" percentage changes (certainly less than 10%), are your safe bet for the approximation to be acceptable.
