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

A54
appendix G The Logarithm Defined as an Integral
Proof of Law 1 Using the first law of logarithms and Equation 10, we have
lnse x e y d − lnse x d 1 lnse y d − x 1 y − lnse x1y d
Since ln is a one-to-one function, it follows that e x e y − e x1y .
Laws 2 and 3 are proved similarly (see Exercises 6 and 7). As we will soon see,
Law 3 actually holds when r is any real number.
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We now prove the differentiation formula for e x .
12
d
dx se x d − e x
Proof The function y − e x is differentiable because it is the inverse function of
y − ln x, which we know is differentiable with nonzero derivative. To find its derivative,
we use the inverse function method. Let y − e x . Then ln y − x and, differentiating
this latter equation implicitly with respect to x, we get
1
y
dy
dx − 1
dy
dx − y − e x
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General Exponential Functions
If b . 0 and r is any rational number, then by (9) and (11),
b r − se ln b d r − e r ln b
Therefore, even for irrational numbers x, we define
13 b x − e x ln b
Thus, for instance,
2 s3 − e s3 ln 2 < e 1.20 < 3.32
The function f sxd − b x is called the exponential function with base b. Notice that b x is
positive for all x because e x is positive for all x.
Definition 13 allows us to extend one of the laws of logarithms. We already know that
lnsb r d − r ln b when r is rational. But if we now let r be any real number we have, from
Definition 13,
Thus
ln b r − lnse r ln b d − r ln b
14 ln b r − r ln b for any real number r
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