Page 162 - Calculus Demystified

Transcendental Functions
CHAPTER 6
logarithm: In other words, 149
x 1
natural logarithm = ln x = dt.
1 t
For 0 <x < 1 the value of ln x is the negative of the actual area between the
graph and the x-axis. This is so because the limits of integration, x and 1, occur in
x
reverse order: ln x = (1/t) dt with x< 1.
1

Fig. 6.1
Notice the following simple properties of ln x which can be determined from
looking at Fig. 6.1:
(i) When x> 1, ln x> 0 (after all, ln x is an area).
(ii) When x = 1, ln x = 0.
(iii) When 0 <x < 1, ln x< 0

x 1 1 1
since dt =− dt < 0 .
1 t x t
(iv) If 0 <x 1 <x 2 then ln x 1 < ln x 2 .
We already know that the logarithm satisfies the multiplicative property. By
applying this property repeatedly, we obtain that: If x> 0 and n is any integer then
n
ln(x ) = n · ln x.
A companion result is the division rule: If a and b are positive numbers then
a

ln = ln a − ln b.
b
EXAMPLE 6.1
Simplify the expression
3 2
a · b
A = ln .
c −4 · d