Page 177 - Calculus Demystified

Transcendental Functions
CHAPTER 6
164
We will be able to do calculations much more easily if we learn some simple
properties of logarithms and exponentials.
If a> 0 and b> 0 then
a (log b) = b.
a
If a> 0 and b ∈ R is arbitrary then
b
log (a ) = b.
a
If a> 0,b > 0, and c> 0 then

(i) log (b · c) = log b + log c
a
a
a
(ii) log (b/c) = log b − log c
a a a
log b
(iii) log b = c
a
log a
c
1
(iv) log b =
a
log a
b
(v) log 1 = 0
a
(vi) log a = 1
a
α
(vii) For any exponent α, log (b ) = α · (log b)
a a
We next give several examples to familiarize you with logarithmic and
exponential operations.
EXAMPLE 6.21
Simplify the expression
4
log 81 − 5 · log 8 − 3 · ln(e ).
3
2
SOLUTION
The expression equals
4 3 4
log (3 )−5·log (2 )−3·lne = 4·log 3−5·[3·log 2]−3·[4·lne]
3
3
2
2
= 4·1−5·3·1−3·4·1=−23.
You Try It: What does log 5 mean in terms of natural logarithms?
3
EXAMPLE 6.22
Solve the equation
4
x 3x
5 · 2 =
7 x
for the unknown x.