I was studying the linear regression in statics. When learning multiple linear regression, the textbook pointed out that $Y_i=\beta_0\exp(\beta_1X_i)+\epsilon_i $ was a nonlinear regression model. (I found some webpages about it online: https://onlinecourses.science.psu.edu/stat501/node/372 .)
I got confused by the part that, they used a $\log$ transformation to obtain the solution where $\log(Y)=\log(\beta_0)+\log(\beta_1X_i)$ like in the previous webpage and http://www.real-statistics.com/regression/exponential-regression-models/exponential-regression/ and Linear, quadratic and exponential regression .
It looked to me, and as they had pointed out, that, after the $\log$ transformation, the newly obtained variables was clearly a linear regression. Further, the term $\log(\epsilon_i)$ was missing from the expression.
My question was that:
What happened to the term of $\log(\epsilon_i)$? Why we could simply dismissed it?
If $\epsilon_i$ still followed a $N(0,\sigma^2) $ distribution, what's the distribution for $\log(\epsilon_i)$?
Further, $\epsilon_i$ could be negative, and although we could got a value through complex analysis, it would still came with a imaginary part. What happened then?
(We had only learnt linear regression so I wan't so sure if there was some context missing.)