I'm trying to solve an exercise but I find it difficult to interpret.
I have a linear-$\log$ model like this: $y = 1 + 0.55\ln(x) + 3z - 2.2w + \text{error term}$.
I wonder what happens to $\beta_1$ when I substitute $\ln(x)$ with $\ln(2x)$. Am I wrong, or the new $\beta_1$ should be just $\beta_1\frac{1}{2}$?
Thank you in advance.
Your original model is
$$ E[y|x, y, z] = \beta_0 + \beta_1 \ln(x) + \beta_2 z+ \beta_3 w $$ thus, in the original model $$ \frac{\partial}{\partial x}E[y|x, y, z] = \beta_1 \frac{\partial}{\partial x} \ln (x) = \beta_1 /x. $$ In the new model $$ \frac{\partial}{\partial x}E[y|x, y, z] = \beta_1 \frac{\partial}{\partial x} \ln (2x) = \beta_1 \frac{2}{2x} = \beta_1/x. $$ That is, using $\ln(Cx)$, $C\in \mathbb{R}$ does not change the impact of small changes in $x$ on $y$. However, it does add a factor of $\ln(2)$ (as you have noted in the comments) to the hyperplane that your model represents.