Page 175 - Calculus Demystified

162
a b CHAPTER 6 Transcendental Functions
(ii) a b−c =
a c
b c
(iii) (a ) = a b·c
b
(iv) a = d if and only if d 1/b = a (provided b = 0)
0
(v) a = 1
1
(vi) a = a
c
c
c
(vii) (a · d) = a · d .
EXAMPLE 6.17
Simplify each of the expressions
5 −7 · π 4 2 3 4
4 ln 3
(e ) , , (3 · x ) .
5 −3 · π 2
SOLUTION
We calculate:

4 ln 3
(e ) = e 4·ln 3 = (e ln 3 4 4
) = 3 = 81;
5 −7 · π 4 −7−(−3) 4−2 −4 2 1 2
= 5 · π = 5 · π = · π ;
5 −3 · π 2 625
2 4
3 4
12
3 4
8
2
(3 · x ) = (3 ) · (x ) = 3 · x 12 = 6561 · x .
4
You Try It: Simplify the expression ln[e 3x · e −y−5 · 2 ].
EXAMPLE 6.18
Solve the equation
3
8
(x · 5) = 9
for x.
SOLUTION
We have

3
3
3
8
(x · 5) = 9 ⇒ x · 5 = 9 1/8 ⇒ x = 9 1/8 · 5 −1
9 1/24
−1 1/3
1/8
⇒ x = (9 · 5 ) ⇒ x = .
5 1/3
x
You Try It: Solve the equation 4 · 3 2x = 7. [Hint: Take the logarithm of both
sides. See also Example 6.22 below.]