My understanding of $e$ is that it is $\lim_{n \to \infty} (1+\frac{1}{n})^n$. When we are estimating the growth rate, we are trying to find $\lim_{n \to \infty} (1+\frac{r}{n})^n$. If we say $x=\frac{r}{n}$ then the growth equation turns into $(1+\frac{1}{x})^{xr}.$ This looks very similar to $e$ and so it makes sense to me that we use $e^r$ to estimate the growth rate. Also, that we use $ln(x)=r$ makes sense when we are trying to solve for the growth rate, because we are basically solving for $r$ in the equation $e^r=x$.
However, in one of my textbooks, it says that we should estimate the growth rate with $100(\ln(x_1)-\ln(x_2))$ where $x_1$ and $x_2$ are subsequent data points, i.e. $x_1=Sales_{2016}$ and $x_2=Sales_{2015}.$ This does not make sense to me because I thought that $\ln(x_1)$ was solving for $x_1$'s growth rate. Can anyone help me understand this?
From $$100(\ln(x_1)-\ln(x_2)) = r$$ you get $$\frac{x_1}{x_2} = e^{\frac{r}{100}}.$$
My guess is that your textbook is using percentages. Your usual $r$ is the $r/100$ of the book.