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

appendix G The Logarithm Defined as an Integral A55
The general laws of exponents follow from Definition 13 together with the laws of
exponents for e x .
15 Laws of Exponents If x and y are real numbers and a, b . 0, then
1. b x1y − b x b y 2. b x2y − b x yb y 3. sb x d y − b xy 4. sabd x − a x b x
Proof
1. Using Definition 13 and the laws of exponents for e x , we have
3. Using Equation 14 we obtain
b x1y − e sx1yd ln b x ln b 1 y ln b
− e
− e x ln b e y ln b − b x b y
sb x d y − e y lnsb xd − e yx ln b − e xy ln b − b xy
The remaining proofs are left as exercises.
■
y
The differentiation formula for exponential functions is also a consequence of Defi nition
13:
y
16
d
dx sb x d − b x ln b
1
Proof
1
0
x
d
dx sb x d − d
dx se x ln b d − e d x ln b dx sx ln bd − b x ln b
0
x
■
lim b®=0, lim b®=`
x _` x `
FIGURE 8 y − b x , b . 1
y
lim b®=`, lim b®=0
x _` x `
If b . 1, then ln b . 0, so sdydxd b x − b x ln b . 0, which shows that y − b x is
increasing (see Figure 8). If 0 , b , 1, then ln b , 0 and so y − b x is decreasing (see
Figure 9).
General Logarithmic Functions
If b . 0 and b ± 1, then f sxd − b x is a one-to-one function. Its inverse function is called
the logarithmic function with base b and is denoted by log b . Thus
1
17 log b x − y &? b y − x
x
0
x
`
lim b®=`, lim b®=0
x _` x `
In particular, we see that
FIGURE 9 y − b x , 0 , b , 1
log e x − ln x
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