The number \(e\), Euler’s number, 2.7182818284…, is special. It’s special because:

\[\frac{d}{dx} e^x = e^x\]

But that’s one of the only reasons it’s special.

It seems to pop up all the time, population growth, radioactive decay, random events and continuous compound interest. I’ve found that I keep needing to remind myself that \(e\) isn’t like \(\pi\) or or the golden ratio \(\phi\), it doesn’t just happen to pop up in nature. It’s there because we put it there, we use \(e\) in mathamatical models because it’s easy to manipulate.

Consider a simple population growth model:

\[N(t) = N_0 e^{at}\]

Why is \(e\) there? Wouldn’t it make more sense to use something like \(2^x\), meaning the population doubles every time unit? Or some other base to better reflect the behaviour of the population?

Well actually they are equivalent. Any exponential function \(a^x\) can be written as \(e^{bx}\) for some choice of \(b\):

\[\begin{align} a^x &= e^{bx}\\ \log_a \left(a^x\right) &= \log_a \left(e^{bx}\right)\\ x &= bx \log_a \left(e\right)\\ 1 &= b \log_a \left(e\right)\\ \frac{1}{\log_a \left(e\right)} &= b \end{align}\]

So:

\[a^x = e^{\frac{1}{\log_a(e)}x}\]

Applying \(\log_e\) or \(\ln\) to both sides gives the change of base formula for logs:

\[\log_c(y) = \frac{\log_d(y)}{\log_d(c)}\]