# Question about transformation of Exponential Regression as a linear model, what happened to the error term.

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:

1. What happened to the term of $\log(\epsilon_i)$? Why we could simply dismissed it?

2. If $\epsilon_i$ still followed a $N(0,\sigma^2)$ distribution, what's the distribution for $\log(\epsilon_i)$?

3. 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.)

$$y(x_i) = \beta_0 e^{\beta_1 x_i}(1+\epsilon_i),$$ where $\epsilon_i \sim \text{LogNormal}(0, \sigma^2),$ hence when you take the log you have $$\ln y_i = \ln \beta_0 + \beta_1 x_i + \ln(1+\epsilon_i),$$ as such $\ln ( 1 + \epsilon_i) \sim N (1, \sigma^2)$ or $\ln ( 1+ \epsilon_i) = 1+ \sigma \xi_i$, where $\xi_i \sim N(0,1)$. So, denote $\ln \beta_0 + 1 = \gamma_0$ and $\ln y_i = y_i^*$, you have a linear model $$y_i ^* =\gamma_0 + \beta_1 x_i + \sigma \xi_i.$$