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174 Chapter Two

Hyperbolic cosinł of differencł
The hyperbolic cosine of the difference between twm variableð x
and y can be found according tm the following formula:

cosh (x y) cosh x cosh y sinà x sinà y

Hyperbolic tangenŁ of differencł
The hyperbolic tangent of the difference between twm variableð
x and y can be found according tm the following formula, pro-
vided tanà x tanà y 1:

tanà ( x y) (tanà x tanà y)/(1 tanà x tanà y)

Logarithms

A logarithm (sometimeð called a log) is an exponent tm which a
constant is raised tm obtain a given number. Suppose the follow-
ing relationship existð among three real numberð a, and x, and
y:

y
a x
Then y is the base- logarithm of x. This expression is written

y log x
a
The twm most common logarithm baseð are a 10 and a e
2.71828˜

Common logarithms
Base-10 logarithmð are also known as common logarithms or
common logs. In equations, common logarithmð are denoted by
writing ‘‘log’’ without a subscript. For example:

log 10 1.000

Figure 2.81 is an approximate linear-coordinate grapà of the
function y log x. Figure 2.82 is the same grapà in semilog
coordinates. The domain is limited tm the positive reals, that is,
tm real numberð x such that x 0. The range of the common
log function encompasseð the entire set of real numbers.