Exponential functions

Put one grain of rice on the first square of a chessboard, double it on every next square, and after only a few squares no sack in the world is big enough. This force sits behind every process that keeps multiplying itself: money earning interest, dividing bacteria, a rumour making the rounds. The mathematics behind it is the exponential function.

The x sits in the exponent

With power functions $x$ sat in the base, as in $y = x^2$. With the exponential function it is the other way round: now $x$ sits up in the exponent. The general form is $y = a \cdot b^x$. Here $a$ is the starting value at $x = 0$, because $b^0 = 1$. The base $b$ is the factor by which the value multiplies at every step. If $b$ is greater than $1$, that fixed factor turns into a curve which grows ever steeper.

Growth and decay

Everything hangs on the base $b$. If $b > 1$ the value grows, at $b = 2$ it doubles with every step. If $b$ lies between $0$ and $1$ the value gets smaller, that is called decay. At $b = 0.5$ it halves each time. That is how the amount of a radioactive material drops, and how the temperature gap between a hot coffee and the room closes. At exactly $b = 1$ nothing happens: times $1$ leaves every value unchanged, the graph is a horizontal line. A positive starting value $a$ only shifts the height, the direction is decided by $b$ alone.

The doubling time

With growth one often asks: how long until the value doubles? This doubling time is always the same length, no matter how large the value already is. If bacteria double every hour, then $200$ becomes $400$ after one hour, $800$ after two hours and $1600$ after three hours. It is exactly this fixed doubling time that makes exponential growth look so modest at first and so overwhelming later.