Powers and logarithms

Repeated multiplication, what the shorthand means once the count stops being a whole number, where the constant e comes from, and the operation that undoes it.

Read in order

  1. Growth by a constant factorRepeated multiplying eventually outruns repeated adding. What must the shorthand mean when its count is not whole, and where does e come from?
  2. The natural logarithmGrowing by a constant factor turns a count into a size. What runs that backwards, and why does running it backwards turn multiplication into addition?