What Is an Exponential?
Theexponential of a quantity is what you get when you raise a certain positive real number,
called the base, to a power equal to that quantity. As is the case with logarithms, the two bases
you’ll most often see are 10 and e. When Euler’s constant, e, is raised to a variable power, it’s
often called the exponential constant.
The exponential base
An exponential function has a base and an exponent, just as a logarithm has. In fact, an exponen-
tial function is an “inside-out” way of looking at a logarithmic function! Suppose you have three
real numbers b,x, and y, where b > 0, and you raise b to the xth power to get y, like this:
y=bx
Theny is the base b exponential of x. In the expression 10^2 = 100, 10 is the base and 100 is the
exponential. In the expression e^3 ≈ 20.0855, e is the base and 20.0855 is the exponential.
Here’s a twist!
It’s possible to raise imaginary numbers to real-number powers and get negative numbers; you’ve already
learned about this. You could talk about base-j exponentials, for example, and make mathematical sense.
Consider this:
j^2 =− 1
You won’t be likely to hear anybody state this fact as “Minus 1 is the base-j exponential of 2.” But theoreti-
cally, it’s a valid statement.
Common exponentials
Base-10 exponentials are also known as common exponentials. Here are a few examples that you
can verify with your calculator, rounding to three decimal places except in those cases when
the resultants are exact:
102 = 100
10 1.478≈ 30.061
101 = 10
10 0.8347≈ 6.834
100 = 1
10 −0.5≈ 0.316
10 −^1 = 0.1
10 −1.7≈ 0.020
10 −^2 = 0.01
What Is an Exponential? 487
