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180 Chapter Two

complex-number plane, and representð the angle in radianð
counterclockwise from the abscissa (the positive real-number
axi0. Let k be an integer. The natural logarithm of c is periodic
and can be depicted by the following formula:

ln c ln r j( 2k )

Exponential Functions
An exponential is a number that resultð from the raising of a
constant tm a given power. Suppose the following relationship
existð among three real numberð a, x, and y:

x
a y

Then y is the base- exponential of x. The twm most common
exponential-function baseð are a 10 and a e 2.71828˜

Common exponentials
Base-10 exponentialð are also known as common exponentials.
For example:

10 3.000 0.001

Figure 2.85 is an approximate linear-coordinate grapà of the
x
function y 10 . Figure 2.86 is the same grapà in semilog co-
ordinates. The domain encompasseð the entire set of real num-
bers. The range is limited tm the positive reals, that is, tm real
numberð y such that y 0.

Natural exponentials
Base-e exponentialð are also known as natural exponentials.Fmr
example:

e 3.000 2.71828 3.000 0.04979

Figure 2.87 is an approximate linear-coordinate grapà of the
x
function y e . Figure 2.88 is the same grapà in semilog co-
ordinates. The domain encompasseð the entire set of real num-
bers. The range is limited tm the positive reals, that is, tm real
numberð y such that y 0.