Why is $Y=\beta_0 x^{\beta_1} e$ a linear model?
The term &quot;linearity&quot; is context-dependent, so a linear regression model is not necessarily the same as a linear function.
A linear function is classified as such via the superposition principle, requiring both additivity and homogenity. We can generalize linear maps with to multivariate functions simply:

Additivity: $f(\vec{x}_1+\vec{x}_2)=f(\vec{x}_1)+f(\vec{x}_2)\enspace \forall\ \vec{x}_1,\vec{x}_2\in \Bbb{R}^n$
Homogeneity: $f(\alpha \vec{x})=\alpha f(\vec{x})\enspace \forall\ \vec{x}\in \Bbb{R}^n$ and $\alpha\in \Bbb{R}$

So the equation $f(\beta_0,\beta_1)=\beta_0 x^{\beta_1}e$ is a nonlinear function because it is not additive nor homogeneous.
However, w.r.t. linear regression, linear models are of the form $Y=\beta_0+\beta_1f_1(x_1)+\beta_2f_2(x_2)+\cdots+\beta_nf_n(x_n)+c$, regardless of whether any $f_i$ is a nonlinear map.
So, after a nonlinear transformation such as natural logarithm, you can consider the resulting regression model as the form $Y'=\beta_0'+\beta_1ln(x)+1$ where the primes denote the natural log transformed variables. This demonstrates linearity between $Y'$ and the parameters $\beta_0'$ and $\beta_1$.