Question

could someone explain to me why we use e in the formula A=Pe^rt?

Answer 1

it's a bit of a magic number, especially when it comes to calculus.

so \( A=Pe^{rt} \implies \dot A = rPe^{rt}\)

Answer 2

Yeah, we use it cause it has this nice property, really we can use any number there though.

Since \(r\) is just an arbitrary number, we could replace it with another arbitrary number: \(r =s \log_e (b)\), then you'll get:
\[A = Pe^{rt} = Pe^{(s \log_e(b) )t} = P b^{st} \]

Since \(e^{\log_e b} = b\).

You can solve for s just like you solved for r, it doesn't matter because when you actually solve for it, these things will sort themselves out, but like @IrishBoy123 it's usually more convenient to use base e.