Conversely, what attributes of the physical universe do attributes of the natural logarithm model?
For example, the exponential function based on the natural logarithm has itself as its derivative. That seems to model aspects of the physical universe.
What are other math-physics correspondences for the natural logarithm?
The natural log (and exponential) arise in any process where the rate of change is proportional to the current value. This often occurs in nature. Examples include reproduction and spread of disease. The calculus begins with dx/dt=a*x and x(t=0)=b, which leads to x=b*exp(a*t) or t=ln(x/b)/a.
Borrowing from Stefanie Reichert: The constant e appears practically everywhere in science: popping up in the definition of the standard normal distribution; allowing us to decompose a time-dependent signal into its frequencies via Fourier transformation; telling us how to calculate the half-life of radioactive elements; playing a crucial role in the growth of bacteria; and governing temperature-activated chemical reactions.
Additionally mathematical physics radioactive decay process has exponential or half-life as logarithmic dependence, example, carbon-14 to know age of a tree.
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