James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

appendix G The Logarithm Defined as an Integral A53
y
1
y=exp x
y=x
y=ln x
0 1
x
y − x. (See Figure 6.) The domain of exp is the range of ln, that is, s2`, `d; the range
of exp is the domain of ln, that is, s0, `d.
If r is any rational number, then the third law of logarithms gives
Therefore, by (6),
lnse r d − r ln e − r
expsrd − e r
Thus expsxd − e x whenever x is a rational number. This leads us to define e x , even for
irrational values of x, by the equation
FIGURE 6
e x − expsxd
In other words, for the reasons given, we define e x to be the inverse of the function ln x.
In this notation (6) becomes
8 e x − y &? ln y − x
and the cancellation equations (7) become
9 e ln x − x x . 0
10 lnse x d − x for all x
y
y=´
The natural exponential function f sxd − e x is one of the most frequently occurring
functions in calculus and its applications, so it is important to be familiar with its graph
(Figure 7) and its properties (which follow from the fact that it is the inverse of the natural
logarithmic function).
FIGURE 7
The natural exponential function
1
0
1
x
Properties of the Exponential Function The exponential function f sxd − e x is
an increasing continuous function with domain R and range s0, `d. Thus e x . 0
for all x. Also
lim e x − 0
x l2`
lim e x − `
x l `
So the x-axis is a horizontal asymptote of f sxd − e x .
We now verify that f has the other properties expected of an exponential function.
11 Laws of Exponents If x and y are real numbers and r is rational, then
1. e x1y − e x e y 2. e x2y − e x
e y
3. se x d r − e rx
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