3 3.0 a n a n ( ) () a m a n = a m+n () (a m ) n = a mn (3) (ab) n = a n b n (4) a m a n = a m n ( m > n ) m n 4 ( ) 55
3. (n ) a n n a n a n 3 4 = 8 8 3 ( 3) 4 = 8 3 8 ( ) ( ) 3 = 8 8 ( ) 3 n n 4 n n (n ) () n a a n () n a > 0 a n a < 0 a n y = x n 39 y = x n n 553
a n x n = a n n a n n 63 () 8 () 6 (3) 8 (4) 3 n n a ( n a ) () n a n n a () n a > 0 a n n a n a > 0 a n n a n a > 0 a n ± n a a a 3 5 = 5 4 8 = 3 5 43 = 3 ( ) 5 43 43 43 = 3 5 64 () 4 6 () 6 79 (3) 3 64 (4) 3 5 554
3. ( ) a > 0, b > 0 m, n () n a n b = n ab, () ( n a) m = n a m m (3) n a = mn a n a n b = n a b () a b = ab a, b n n a ( ) ) () n ( n ) n n ( a b = n ) ( ) n n n a b = ab n ( ) n n ab = ab n a n b = n ab 40 3 3 8 = 3 6 = 6 4 8 4 = 4 6 = 8 ( ) 6 3 6 9 = 9 3 = 6 79 = 3 3 4 = = ( ) 555
3 3 8 = 3 3 3 3 = 3 ( 3) 3 = 3 3 3 3 = 3 3 3 3 3 = 3 = 6 65 () 3 4 9 3 64 3 () 4 (3) ( 6 4 ) 3 4 (4) 3 79 4 a > 0 m, n, p np a mp = n a m 0 3 6 3 8 + 3 3 3 6 6 = 4 4 3 3 6 = 3 3 = 3 3 3 = 3 3 8 3 8 = 4 3 (!) 556
( ) = 3 4 3 + 3 3 = 3 ( ) ( ) 66 () 3 4 3 3 + 3 8 () 3 6 + 3 3 8 3.3 3.3. ( ) 4 0 a 0 a m a n = a m+n m =, n = 0 aa 0 = a a a 0 = a m a n = a m+n m = n a n a n = a 0 = a n = a n 0 0 557
( ) a 0 n a n = a n a 0 = n n a = a a a ( ) a 0, b 0 m, n () a m a n = a m+n () (a m ) n = a mn (3) (ab) n = a n b n () m n a 0, b 0 55 a 0 () 55 (4) a m a n = a m a n = am a n () (4) m > n 4 m > n (I) m 0, n 0 (II) m 0, n 0 (III) m 0, n 0 (I) n 0 n n = n 558
() m n a m a n = a m a n = a m a n m n a m = a m n a m+n a n a m a n = a m+n m < n () (a m ) n = (a m ) n = = (a m ) n a mn = a mn = a m ( n ) = a mn (3) (ab) n = (ab) n = (ab) n = a n b n = a n b n = a n b n = a n b n 43 m, n () x 5 x 4 = x 9 () (x ) 3 = x 6 (3) (a b ) 3 = a 6 b 3 ( ) 67 () a a 5 () (a ) 4 (3) (a b 3 ) (4) x 5 x x 4 (5) (a ) 3 a 6 (6) (a ) 5 a 68 () 4 () ( 5) 0 (3) 3 4 9 3 (4) 3 8 (5) {( ) 3 } 5 (6) (45 0 ) 30 (45 30 ) 0 559
3.3. a > 0 (a m ) n = a mn m m m = q p p, q ( ) a q p q p = a p p = a q a q p a q p a q p = p a q ( ) a > 0 q p a q p = p a q a p = p a ( ) 6 = 3 6 3 a 6 a 3 a 6 a 3? 556 np a mp = n a m a 6 = 6 a = 3 a 3 = 3 a = 3 a = a 3 560
a 6 a 3 ( ) mp np = m n 44 ( ) a > 0, b > 0 r, s () a r a s = a r+s () (a r ) s = a rs (3) (ab) r = a r b r a > 0, b > 0 r, s r = n m, s = n a r a s = a n m b n m m = m a n m a n = mm a m n mm a mn = mm a m n a mn = mm a m n+mn = a m n+mn mm = a n m + n m = a r+s 56
45 () a 3 a = a 3 + = a 5 6 () 3 = 3 + = 5 6 ( ) (3) a 4 = a 4 = a ( ) (4) 9 4 = 9 4 = 9 = 3 (5) (a b 3 ) 3 = a 3 (b 3 ) 3 = a 3 b (6) 54 3 = ( 3 3 ) 3 = 3 (3 3 ) 3 = 3 3 = 3 3 ( ) 69 () a a 3 4 a 4 () 6 6 3 4 6 4 (3) (a 3 b) (ab ) 3 (4) 4 8 3 3.3.3 a 4 ( ) ( ) α {a n } lim a n = α n 56
an {a n } {a n } a = lim n aa n a a n a n {a a n } {an } ( ) lim a n = = lim b n {b n } n n lim n aa n = lim n ab n ( ) ( ) a > 0, b > 0 x, y () a x a y = a x+y () (a x ) y = a xy (3) (ab) x = a x b x y = a x 3.4 ( ) 563
() a > x > y a x > a y () 0 < a < x > y a x < a y a a 0 x, y a > 0 < m < a m a a > a a > a a a 3 > a < a < a < a 3 < < a m < a m ( ) 46 a > < a m 0 < m < m a m < a m < a m a m a m < a m < n a n < m, n m < n (m, n ) a m < a n 47 564
r, s ( ) a > 0, b > 0 a > b a n > b n 8 ( ) ( ) a > 0, b > 0 a > b n a > n b a = ( n a) n 48 r = n m, s = n m ( ) r > s n m > n m m m m n > mn m n, mn a m n > a mn mm a n m > a n m 49 50 0 < a < 03 3, 3 9, 5 7 3 x 3 = 3, 3 9 = 3 3, 5 3 7 = 3 5 3 3 = 3 3 9 = 3 5 3 7 = 3 5 565, 3,
3 > < 3 5 < 3 3 3 < 3 5 < 3 3 3 < 5 7 < 3 9 ( ) ( ) 3 > 70 () 3 4,, 4 8 () 0., 0., 0. 3 04 9 x+3 < 79 9 x+3 9 9 = 3 3 79 = 9 3 9 x+3 < 9 3 9 > () x + 3 < 3 x 9 x+3 < 9 3 9 > x + 3 < 3 x < 0 ( ) ( ) 566
7 ( ) () 4 3x > 56 () x+ 8 ( ) a, a > 0 x = y a x = a y = ) a x a y a x > a y a x < a y a x > a y a > x > y 0 < a < x < y x y a x < a y =) x y x > y x < y = ) a x a y 5 05 4 x+ = 3 6 4 3 6 = 4 3 4 x+ = 4 3 x + = 3 3 6 = 4 3 4 x+ = 4 3 x + = 3 x = 3 ( ) ( ) 567
7 () x+ = 6 () 4 3x+ = 6 3.5 ( ) a, a > 0 x a x a, a > 0 M a r = M r ( ) a, a > 0 M a r = M r r log a M a M M log a M M > 0 a, a > 0, M > 0 r = log a M M = a r () 0 3 = 000 log 0 000 = 3 ( ) () 3 = 7 log 7 = 3 3 3 568
( ) 73 r = log a M () 3 3 = 7 () (0.) = 00 (3) 3 8 = (4) ( ) 6 = 8 (5) 4 = 4 (6) 9 0 = 06 x a () log 3 x = 4 () log a 8 = 3 () 3 4 = x () a 3 = 8 () 3 4 = x x = 8 ( ) () a 3 = 8 a = ( ) ( ) 74 x a () log 3 x = () log x = 3 (3) log a 64 = 6 (4) log a 3 = 07 () log 5 65 () log 3 () log 5 65 = x 5 x = 65 65 5 () log 3 = x ( ) x = 3 569
x? = ( ) x x = 5 x () log 5 65 = x 5 x = 5 4 x = 4 ( ) () log 3 = x x = 5 x = 5 ( ) ( ) 75 () log 64 () log 5 5 3.6 3.6. a 0 =, a = a ( )) () log a = 0 () log a a = 570
a m a n = a m+n, a m a n = a m n, (a m ) n = a mn ( )) M > 0, N > 0, a > 0, a, r () log a MN = log a M + log a N () log M a N = log a M log a N (3) log a M r = r log a M M = a m, N = a n MN = a m a n = a m+n log a MN = m + n m = log a M, n = log a N log a MN = log a M + log a N 5 ( ) log 3 log 5 65 = log 5 5 4 = 4 log 5 5 = 4 = 4 ( ) log 4 3 + log 6 = log ( 4 3 6 ) = log 8 = log 3 = 3 log = 3 ( ) 57
log 4 log4 3 = log 4 ( ) log 4 3 = log 4 log 4 3 = log 4 3 = log 4 4 = ( ) 76 () log 64 () log 0 8 log 4 0 5 (3) log 5 + log 40 (4) log 5 + log 5 3 log 5 6 3.6. ( ) a, b M > 0 log a M = log b M log b a r = log a M a r = M b log b a r = log b M r log b a = log b M r = log a M log a M log b a = log b M a log b a 0 log b a log a M = log b M log b a 57
a b a b a M log 9 7 9 7 3 3 log 9 7 = log 3 7 log 3 9 = 3 ( ) log 3 log 3 = log 3 log = 5 = 5 ( ) 77 () log 4 3 () log 7 9 08 log 3 log 3 8 log 3 log 3 8 log 3 log 3 log 3 8 = log 3 log 8 log 3 = log 8 = 3 ( ) ( ) 573
53 3 78 () log 9 5 log 5 9 () log 3 log 3 4 log 4 8 () ( ) log a b = log b a 3.7 ( ) () a > x > y a x > a y () 0 < a < x > y a x < a y (3) a, a > 0 x = y a x = a y ( ) () a > x > y log a x > log a y () 0 < a < x > y log a x < log a y (3) a, a > 0 x = y log a x = log a y 54 log 3, log 3 3, log 3.8 3 () 3 <.8 < log 3 3, log 3.8, log 3 574 ( )
log 0.5 0., log 0.5 0.9, log 0.5 0.09 () 0.09 < 0. < 0.9 log 0.5 0.9, log 0.5 0., log 0.5 0.09 ( ) 79 () 3 log 3, log 5 () 3, log 4 4, log 5 3.8 3.8. log a x a a > 0, a log 3 ( ) 0 log 3 = log 0 3 log 0 log 0 3 log 0 log 3 (799 ) log 0.48 = 0.3945 575
.4 8 log 0 48 48 48 =.48 00 =.48 0 log 0 48 = log 0 (.48 00) = log 0.48 + log 0 00 =.3945 log 0 0.48 0.48 =.48 0 log 0 0.48 = log 0 (.48 0 ) = log 0.48 = 0.6055 ( ) 80 4 () log 0 7.34 () log 0 644 (3) log 0 0.0003 log 0 = 0.300, log 0 3 = 0.477 log 3 = log 0 3 log 0 = 0.477 0.300 =.5850 ( ) 8 log 0 = 0.300, log 0 3 = 0.477 () log 0 6 () log 0 (3) log 0 5 ( 5 = 0 ) 576
3.8. ( ) log 0 00, log 0 0000, log 0 0000000 log 0 00 = log 0 0000 = 4 log 0 0000000 = 7 00 3 0000 5 0000000 8 0 n n + x n 0 n x < 0 n 0 n log 0 x < n x n log 0 x < n n x n 09 0 log 0 = 0.300 0 0 log 0 0 = 0 log 0 = 6.00 577
6 log 0 0 < 7 n 7 7 0 0 log 0 0 = 0 log 0 = 6.00 6 log 0 0 < 7 0 7 ( ) ( ) 6 log 0 0 < 7 log 0 0 = 6.00 7 log 0 0! 8 3 0 log 0 3 = 0.477 3.9 3.9. a > 0 x a x y = a x a 578
a a > 0 < a < y = x y O x () y = x x y y (0, ) a x ( x ) x ( ) y = x y O x ( ) () y = x 579
x y y (0, ) a x a = a = a > a = 0 < a < a = y = a x () a > y () 0 < a < y a a O x O x ( ) () () (0, ) (3) x (4) a > 0 < a < 3.9. a > 0 a x log a x y = log a x x > 0 a 580
a > 0 < a < y = log x y O x () y = log x x y y x (, 0) a x 0 y y y = log x y O x () y = log x 58
x y x (, 0) a y a = a = a > a = 0 < a < a = y = log a x () a > y () 0 < a < y O a x O a x ( ) () () (, 0) (3) y (4) a > 0 < a < y = a x y = log a x a > 58
y O x y = x (p, q) y = a x (p, q) y = x (q, p) q = a p p = log a q (q, p) y = log a x y = a x y = log a x y = x 0 < a < y O x 3.0 583
584