Page 163 - Calculus Demystified

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SOLUTION CHAPTER 6 Transcendental Functions
We can write A in simpler terms by using the multiplicative and quotient
properties:
2
3
A = ln(a · b ) − ln(c −4 · d)
2
3
=[ln a + ln(b )]−[ln(c −4 ) + ln d]
=[3ln a + 2 · ln b]−[(−4) · ln c + ln d]
= 3ln a + 2 · ln b + 4 · ln c − ln d.
The last basic property of the logarithm is the reciprocal law: For any x> 0
we have
ln(1/x) =− ln x.

EXAMPLE 6.2
Express ln(1/7) in termsof ln 7. Express ln(9/5) in termsof ln 3 and ln 5.
SOLUTION
We calculate that
ln(1/7) =− ln 7,
2
ln(9/5) = ln 9 − ln 5 = ln 3 − ln 5 = 2ln 3 − ln 5.
2 −3
5
You Try It: Simplify ln(a b /c ).
6.1.2 THE LOGARITHM FUNCTION AND THE
DERIVATIVE

Now you will see why our new definition of logarithm is so convenient. If we want
to differentiate the logarithm function we can apply the Fundamental Theorem of
Calculus:
x
d d 1 1
ln x = dt = .
dx dx 1 t x
More generally,
d 1 du
ln u = .
dx u dx
EXAMPLE 6.3
Calculate
d d 3 d d 5 d
ln(4 + x), ln(x − x), ln(cos x), [(ln x) ], [(ln x) · (cot x)].
dx dx dx dx dx