Natural Logarithms

Natural Logarithms
For any positive number a, the function value is easy to define when x is an
integer or rational number. When x is irrational, the meaning of is not so clear.
Similarly, the definition of the logarithm the inverse function of is not
completely obvious. In this section we use integral calculus to define the natural logarithm
function, for which the number a is a particularly important value. This function allows
us to define and analyze general exponential and logarithmic functions, and
Logarithms originally played important roles in arithmetic computations. Historically,
considerable labor went into producing long tables of logarithms, correct to five, eight, or
even more, decimal places of accuracy. Prior to the modern age of electronic calculators
and computers, every engineer owned slide rules marked with logarithmic scales. Calculations
with logarithms made possible the great seventeenth-century advances in offshore
navigation and celestial mechanics. Today we know such calculations are done using
calculators or computers, but the properties and numerous applications of logarithms are
as important as ever.
Definition of the Natural Logarithm Function
One solid approach to defining and understanding logarithms begins with a study of the
natural logarithm function defined as an integral through the Fundamental Theorem of
Calculus. While this approach may seem indirect, it enables us to derive quickly the familiar
properties of logarithmic and exponential functions. The functions we have studied
so far were analyzed using the techniques of calculus, but here we do something more
fundamental. We use calculus for the very definition of the logarithmic and exponential
functions.
The natural logarithm of a positive number x, written as ln x, is the value of an
integral.
y = loga x.
y = ax
loga x, ƒsxd = ax ,
ax
ƒsxd = ax
7.2
DEFINITION The Natural Logarithm Function
ln x = L
x
1
1t
dt, x 7 0
If then ln x is the area under the curve from to
(Figure 7.9). For ln x gives the negative of the area under the curve from x to
1. The function is not defined for From the Zero Width Interval Rule for definite
integrals, we also have
ln 1 = L
1
1
1t
dt = 0.
x … 0.
0 6 x 6 1,