Page 62 - Calculus Demystified

Basics
CHAPTER 1

Fig. 1.57 49

1.9 A Few Words About Logarithms

andExponentials

We will give a more thorough treatment of the logarithm and exponential functions
in Chapter 6. For the moment we record a few simple facts so that we may use
these functions in the sections that immediately follow.
The logarithm is a function that is characterized by the property that
log(x · y) = log x + log y.

It follows from this property that
log(x/y) = log x − log y
and
n
log(x ) = n · log x.
It is useful to think of log b as the power to which we raise a to get b, for any
a
a, b > 0. For example, log 8 = 3 and log (1/27) =−3. This introduces the idea
3
2
of the logarithm to a base.
You Try It: Calculate log 125, log (1/81), log 16.
5
3
2
The most important base for the logarithm is Euler’s number e ≈ 2.71828 ... .
Then we write ln x = log x. For the moment we take the logarithm to the base e,or
e
the natural logarithm, to be given. It is characterized among all logarithm functions
by the fact that its graph has tangent line with slope 1 at x = 1. See Fig. 1.58. Then
we set
ln x
log x = .
a
ln a
Note that this formula gives immediately that log x = ln x, once we accept that
e
log e = 1.
e