F.5 指數函數與對數函數

1.
43x x 2-2x x 16x / 82x
= (22)3x x (2)-2x x (24)x / (23)2x
= (2)6x x (2)-2x x (2)4x / (2)6x
= (2)6x-2x+4x-6x
= 22x


2.
(2n+4 - 2n+1) / (2n+4 + 2n+1)
= [2n+1(23 - 1)] / [2n+1(23 + 1)]
= (8 - 1)/(8 + 1)
= 7/9


3.
(2xy2)-1(2x-1y-3)2
= (2-1x-1y-2)(22x-2y-6)
= 2-1+2x-1-2y-2-6
= 2x-3y-8
= 2/x3y8


4.
a-2 - b-2
= (1/a)2 - (1/b)2
= [(1/a) + (1/b)][(1/a) - (1/b)]
= [(b/ab) + (a/ab)][(b/ab) - (a/ab)]
= [(b + a)/ab][(b - a)/ab]
= (b + a)(b - a)/a2b2
= -(a + b)(a - b)/a2b2

所以 (a - b) / (a-2 - b-2)
= (a - b) / [-(a + b)(a - b)/a2b2]
= -(a - b) x [a2b2/(a + b)(a - b)]
= -a2b2/(a + b)

(1) (4^3x ) (2^-2x) (16^x) /(8^2x)
= (2^6x) (2^-2x) (2^4x) (2^-6x)
= 2^(6x-2x+4x-6x)
=2^2x
(2) [2^(n+4) - 2^(n+1)] / [2^(n+4) + 2^(n+1)]
= [16(2^n) - 2(2^n)] / [16(2^n) + 2(2^n)]
= 14(2^n) / 16(2^n)
= 7/8
(3) (2xy^2)^-1 (2x^-1 y^-3)^2
= (2^-1) (x^-1) (y^-2) (2^2) (x^-2) (y^-6)
= 2x^(-1-2) y^(-2-6)
= 2(x^-3)(y^-8)
(4) (a-b) / (a^-2 - b^-2)
= (a-b) / (1/a^2 - 1/b^2)
= (a-b) / [(b^2 - a^2)/a^2 b^2]
= a^2 b^2 (a-b) / [(a-b)(a+b) ]
= a^2 b^2 / (a+b)

2009-08-15 07:56:17 補充：
修正答案
(2) [2^(n+4) - 2^(n+1)] / [2^(n+4) + 2^(n+1)]

= [16(2^n) - 2(2^n)] / [16(2^n) + 2(2^n)]

= 14(2^n) / 18(2^n)

= 7/9

(4) (a-b) / (a^-2 - b^-2)

= (a-b) / (1/a^2 - 1/b^2)

= (a-b) / [(b^2 - a^2)/a^2 b^2]

= -a^2 b^2 (a-b) / [(a-b)(a+b) ]

= -a^2 b^2 / (a+b)