[summary: The exponent base $~$b$~$ of a number $~$x$~$, written $~$b^x,$~$ is what you get when you multiply 1 by $~$b$~$, $~$x$~$ times. For example, the exponent base 10 of 3 is $~$10^3$~$ and pronounced "10 to the power 3" is 1000, because $~$10 \cdot 10 \cdot 10 = 1000$~$. Similarly, $~$2^4=16,$~$ because $~$2 \cdot 2 \cdot 2 \cdot 2 = 16.$~$
The input $~$x$~$ may be fractional. For example, $~$10^{1/2}$~$ is the number you get when you multiply 1 by 10 half a time, where "multiplying by 10 half a time" means multiplying by a number $~$n$~$ such that if you multiplied by $~$n$~$ twice, you would have multiplied by 10 once. In this case, $~$n \approx 3.16,$~$ because then $~$n \cdot n \approx 10.$~$]
[todo: Add numerous other summaries. See, e.g., the summaries at /p/logarithm.] [todo: I suggest making the main page for exponential mirror the main page at /p/logarithm.]
An exponential is a Function that can be represented by some [-constant] taken to the [power_mathematics power] of a [-variable]. The name comes from the fact that the variable is the exponent of the expression.
Exponential Growth
Exponentials are most useful in describing growth patterns where the growth rate is proportional to the amount of the thing that's growing. They can be represented by the formula: $~$f(x) = c \times a^x$~$, where $~$c$~$ is the starting value and $~$a$~$ is the growth factor.
The classic example of exponential growth is [-compound_interest]. If you have \$100 in a bank account that gives you 2% interest every year, then every year your money is multiplied by $~$1.02$~$. This means you can represent your account balance as $~$f(x) = 100 \times 1.02^x$~$, where $~$x$~$ is the number of years your money has been in the account.
Another example is a dividing cell. If one cell is placed into an infinite culture and splits once every hour, the number of cells in the culture after $~$x$~$ hours is $~$f(x) = 1 \times 2^x$~$ (assuming none of the cells die).
Recursive definition
We mentioned earlier that in an exponential function, the growth rate is proportional to the amount of the thing that's growing. In the compound interest example, we can write each value in terms of the previous value: $~$f(x) = f(x-1) \times 1.02$~$. Therefore, the amount of growth at every step can be represented as $~$\Delta f(x) = f(x+1) - f(x) = 0.02 \times f(x)$~$.
This makes exponential growth a memoryless growth function, as the growth rate depends only on current information. Compare this to [-simple_interest], where the interest only grows at a percentage of the initial value. Then we would have to write: $~$f(x) = f(x-1) + 0.02 \times f(0)$~$, and because $~$f(0)$~$ is constant while $~$f(x)$~$ continues to grow, we cannot express the growth rate in terms of only current information — we have to keep "in memory" the initial balance to calculate the growth rate.