I'm wondering why it's slightly wrong, or simply an approximation, to attempt solve a differential equation-problem with a normal exponential function without e in the base.
Here's an example. If a population (y) of initially 3.6 million increases by 0.8% per year (x), why is the correct solution to the related D.E. y'=0.008y, namely y=3.6e^(0.008x), only slightly different than using the exponential function y=3.6*1.008^x?
I know that e^0.008 (which is approx. 1.0080321) is slightly different from exactly 1.008, so I can explain it by just seeing there's a mathematical difference, but what am I doing wrong when I use exactly 1.008 as the base? What's the intuition between the slight error and the actual correct?