# Exponentials and logarithms

*A narrative-style introduction to exponentials and logarithms…*

→ [preprint] ←

At least in most of the calculus texts I have seen, the natural logarithm is introduced in an *ex cathedra* fashion, as if somehow God descended from the Heavens above, and decreed that \(\log x = \int_1^x 1/t \, dt\)—and the exponential function, again by divine decree, is to be the inverse of \(\log\). The fact that I have no wish to usurp on His authority notwithstanding, here I provide another way to introduce both functions, starting with a concrete problem: how to compute the derivative of function \(a^x\). I hope it makes for an clear and enlightening reading.

**April 10, 2024.** *Got feedback? Great, email is your friend!*