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# Exponential growth

In mathematics, a quantity that grows exponentially is one that grows at a rate proportional to its size. This does not mean merely that for any exponentially growing quantity, the larger the quantity gets, the faster it grows. It implies that the relationship between the size of the dependent variable and its rate of growth is governed by a strict law, of the simplest kind: direct proportion. It is proved in calculus that this law requires that the quantity is given by the exponential function, if we use the correct time scale. This explains the name.

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## Intuition

The phrase exponential growth is often used in nontechnical contexts to mean merely surprisingly fast growth. In a strictly mathematical sense, though, exponential growth has a precise meaning which doesn't necessarily mean that growth will happen quickly. In fact, a population can grow exponentially but at a very slow absolute rate (as when money in a bank account earns a very low interest rate, for instance), and can grow surprisingly fast without growing exponentially. And some functions, such as the logistic function, approximate exponential growth over only part of their range. The "technical details" section below/subexp explains exactly what is required for a function to exhibit true exponential growth.

But the general principle behind exponential growth is that the larger a number gets, the faster it grows. Any exponentially growing number will eventually grow larger than any other number which grows at only a constant rate for the same amount of time (and will also grow larger than any function which grows only subexponentially). This is demonstrated by the classic riddle in which a child is offered two choices for an increasing weekly allowance: the first option begins at 1 cent and doubles each week, while the second option begins at $1 and increases by$1 each week. Although the second option, growing at a constant rate of $1/week, pays more in the short run, the first option eventually grows much larger: Week: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Option 1: 1c, 2c, 4c, 8c, 16c, 32c, 64c,$1.28, $2.56,$5.12, $10.24,$20.48, $40.96,$81.92, $163.84,$327.68
Option 2:$1,$2, $3,$4,  $5,$6,  $7,$8,    $9,$10,    $11,$12,    $13,$14,     $15,$16


We can describe these cases mathematically. In the first case, the allowance at week n is 2n cents; thus, at week 16 the payout is 216 = 32768c = $327.68. All formulas of the form kn, where k is an unchanging number (eg., 2), and n is the amount of time elapsed, grow exponentially. In the second case, the payout at week n is simply n dollars. The payout grows at a constant rate of$1 per week.

This image shows a slightly more complicated example of an exponential function overtaking subexponential functions:

The red line represents 50x, similar to option 2 in the above example, except increasing by 50 a week instead of 1. Its value is largest until x gets around 7. The green line represents the polynomial x3. Polynomials grow subexponentially, since the exponent (3 in this case) stays constant while the base (x) changes. This function is larger than the other two when x is between about 7 and 9. Then the exponential function 2x takes over and becomes larger than the other two functions for all x greater than about 10.

Anything that grows by the same percentage every year (or every month, day, hour etc.) is growing exponentially. For example, if the average number of offspring of each individual (or couple) in a population remains constant, the rate of growth is proportional to the number of individuals. Such an exponentially growing population grows three times as fast when there are six million individuals as it does when there are two million. Bank accounts with fixed-rate compound interest grow exponentially provided there are no deposits, withdrawals or service charges. Mathematically, the bank account balance for an account starting with s dollars, earning an annual interest rate r and left untouched for n years can be calculated as $s \times (1+r)^n$. So, in an account starting with $1 and earning 5% annually, the account will have $1\times(1+0.05)^1=1.05$ after 1 year, $1\times(1+0.05)^{10}=1.62$ after 10 years, and$131.50 after 100 years. Since the starting balance and rate don't change, the quantity $1\times(1+0.05)=1.05$ can work as the value k in the formula kn given earlier.

## Technical details

Let x be a quantity growing exponentially with respect to time t. By definition, the rate of change dx/dt obeys the differential equation:

$\!\, \frac{dx}{dt} = k x$

where k > 0 is the constant of proportionality (the average number of offspring per individual in the case of the population). (See logistic function for a simple correction of this growth model where k is not constant). The solution to this equation is the exponential function $\!\, x(t)=x_0 e^{kt}$ -- hence the name exponential growth. The constant $\!\, x_0$ is determined by the initial size of the population.

In the long run, exponential growth of any kind will overtake linear growth of any kind (the basis of the Malthusian catastrophe) as well as any polynomial growth, i.e., for all α:

$\lim_{x\rightarrow\infty} {x^\alpha \over Ce^x} =0$

There is a whole hierarchy of conceivable growth rates that are slower than exponential and faster than linear (in the long run). Growth rates may also be faster than exponential. The linear and exponential models are merely simple candidates but are those of greatest occurrence in nature.

In the above differential equation, if k < 0, then the quantity experiences exponential decay.