Algebra Know-It-ALL

In the first equation, 2 is the argument (the value on which the function depends), and

100 is the resultant. You would say is as “The common exponential of 2 is equal to 100.” In the

second equation, you would say “The common exponential of 1.478 is approximately 30.061.”

As with logarithms, the arguments in an exponential function need not be whole numbers. They

can even be irrational; you could speak of 10π, for example. (It’s approximately equal to 1,385.)

Natural exponentials

Base-e exponentials are also called natural exponentials. Here are some examples using the

same arguments as above, rounding to three decimal places except for e^0 , which is exactly 1:

e^2 ≈ 7.389

e1.478≈ 4.384

e^1 ≈ 2.718

e0.8347≈ 2.304

e^0 = 1

e−0.5≈ 0.607

e−^1 ≈ 0.368

e−1.7≈ 0.183

e−^2 ≈ 0.135

Logarithms vs. exponentials

The exponential function is the inverse of the logarithm function, and vice-versa. When two

functions are inverses, they “undo” each other, as long as both functions are defined for the

all the arguments of interest. A logarithm can be “undone” by the exponential function of the

same base. The reverse of this is also true: an exponential can be “undone” by the log function

of the same base.

Sometimes, the common exponential of a quantity is called the common antilogarithm

(antilog 10 ) or the common inverse logarithm (log−^1 ) of that number. The natural exponential

of a quantity is sometimes called the natural antilogarithm (antiln) or the natural inverse loga-

rithm (ln−^1 ) of that number.

We can illustrate the relationship between a common log and a common exponential

with two equations. If we let the abbreviation “log” represent the base-10 logarithm, then

log (10x)=x

for any real number x, and

10 (logy)=y

for any positive real number y. A similar pair of equations holds for the natural logarithms. We

can replace 10 with e, and replace “log” with “ln” to get

ln (ex)=x

488 Logarithms and Exponentials