The Number e and the Natural Logarithm

Start with interest compounded once a year, then monthly, then daily, then every second. Each time you compound more often, the end-of-year balance on 1 dollar at 100 percent interest creeps up — but it does not run off to infinity. It stalls out near 2.71828..., a fixed number called e. That ceiling is what you get when growth compounds continuously, every instant, instead of in discrete jumps. The formula for that continuous growth is A = Pe^rt, where P is the starting amount, r is the rate written as a decimal, and t is time.

Because e shows up constantly in continuous growth, it earns its own logarithm. ln(x) just means log base e of x — nothing new, just a shortcut name. It answers the question 'e to what power gives me x?' So ln and e^x undo each other perfectly: ln(e^x) = x and e^{ln x} = x. If you solve e^x = 20 and reach for a calculator's log key, that button is base 10, not base e — you'd need to convert bases. Reach for ln instead and you get x = ln(20) in one step.

Negative rates flip growth into decay without changing the formula's shape. In A = Pe^rt, a negative r makes the exponent shrink as t grows, so the amount drains away instead of piling up — same equation, opposite direction. This ties continuous growth and decay to everything else in the unit: logarithms are still the tool for pulling a variable down out of an exponent, and the log laws still apply to ln exactly as they do to any other base.