Exponential Functions with Base e

Exponential functions with base e are built on a special constant, approximately two point seven one eight. Also, it is irrational and non-repeating, similar to pi.

This base naturally models continuous growth if the exponent is positive, or decay when the exponent is negative.

The general form involves e raised to a variable exponent, multiplied by an initial value.

For example, a cup of coffee cooling from ninety degrees toward room temperature, cooling at a continuous rate of twelve percent per minute, follows this exponential pattern.

By Newton’s Law of Cooling, the coffee's temperature after t minutes is the room temperature plus the difference between the coffee’s initial temperature and room temperature, multiplied by e raised to the power of negative zero point one two t.

The negative exponent shows the coffee cools rapidly at first, then slows as the graph flattens toward room temperature. This clearly illustrates how exponential decay approaches a limit.

Consider another example: the early spread of a virus often follows exponential growth with base e. It starts with a few cases, and the exponential growth formula ensures the cumulative increase is zero at t=0 by calculating only the growth since the start.