Why continuous rate uses a logarithm

A regular growth rate uses the factor 1 + r once per period. Continuous growth uses the model e^(r_c) over one period. Setting those equal gives r_c = \ln(1+r).

How to read the output

The result is a decimal rate, not a percentage label. It tells you the exponent used in a continuous model A = Pe^(r_ct) that matches the same one-period growth factor.

Where this appears

Continuous rates appear in finance, population models, and differential-equation settings where change is modeled at every instant instead of at fixed checkpoints.