70 70 1 / 27
70 1 2 3 4 5 6 2 / 27
70 70 70 X r % = 70 2 r r r 10 72 70 72 70 : 1, 2, 5, 7, 10, 14, 35, 70 72 : 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72 3 / 27
r = 10 70 r = 10 70 1 : X, X 10 = ( X + X 10 ) = 1.1 X 1 1.1 2 : 1.1 1.1 = 1.21 3 : 1.21 1.1 = 1.331 1.33 ( 3 ) 4 : 1.33 1.1 1.46, 5 : 1.46 1.1 1.61 6 : 1.61 1.1 1.77, 7 : 1.77 1.1 1.95 8 : 1.95 1.1 2.15 70 : 70 10 = 7 4 / 27
r = 7 70 r = 7 70 1 : X, X 7 1 1.07 2 : 1.07 1.07 1.14, 3 : 1.14 1.07 1.22 4 : 1.22 1.07 1.31, 5 : 1.31 1.07 1.40 6 : 1.40 1.07 1.50, 7 : 1.50 1.07 1.61 8 : 1.61 1.07 1.72, 9 : 1.72 1.07 1.84 10 : 1.84 1.07 1.97 11 : 1.97 1.07 2.11 70 : 70 = 10 7 5 / 27
r 70 r 1 : X, X r ( 1 1 + r ) ( 2 : 1 + r ) ( 1 + r ) ( = 1 + r ) 2 ( n 1 + r ) n 2 ( 1 + r ) n = 2 n, 6 / 27
70 a: 1 : x a x x : n = 1, 2, 3,... a n a n a 1 = a, a 2 = a a, a 3 = a a a,... m, n = 1, 2, 3,... a m+n = a m a n, (a m ) n = a mn 7 / 27
70 : a 0 = 1, a n = 1 a n (n = 1, 2, 3,... ) : m = 1, 2, 3,..., n = 0, ±1, ±2, ±3,... m a n a n m : x x a x x, y a x+y = a x a y, (a x ) y = a xy 8 / 27
0 < a < 1 70 0 < a < 1 y = a x () y 1 a O 1 y = a x x 9 / 27
a > 1 70 a > 1 y = a x () y y = a x a 1 O 1 x 10 / 27
(logarithmic function) 70 a: 1 x: x y x = a y y = log a x x log a x a x, y log a (xy) = log a x + log a y, log a x y = y log a x 11 / 27
1 70 x = a u, y = a v u = log a x, v = log a y xy = a u a v = a u+v ( ) xy = a u+v u + v = log a (xy) log a (xy) = log a x + log a y 12 / 27
2 70 x = a u u = log a x x y = (a u ) y = a uy ( ) x y = a yu yu = log a x y log a x y = y log a x 13 / 27
0 < a < 1 70 0 < a < 1 y = log a x y 1 O a 1 x y = log a x 14 / 27
a > 1 70 a > 1 y = log a x y 1 y = log a x O 1 a x 15 / 27
70 70 X r % n ( 1 + r ) n = 2 n a 1 a : ( log a 1 + r ) n = loga 2 16 / 27
70 70 ( log a 1 + r ) n ( = loga 1 + r ) r = log a { ( 1 + r rn ) } rn r ( : a xy = (a x ) y ) = rn ( log a 1 + r ) r ( : log a x y = y log a x) rn ( log a 1 + r ) r = log a 2 17 / 27
70 70 x = r ( log a 1 + r ) r = log a (1 + x) 1 x x = r 0 x = 1: (1 + 1) 1 = 2, x = 1 5 : ( 1 + 1 5 5) 2.488 x = 1 10 : ( 1 + 1 10 ) 10 2.594, x = 1 : ( 1 + 1 ) 2.705 x = 1 0 : ( 1 + 0) 1 0 2.717 x = 1 00 : ( 1 + 00) 1 00 2.718 x = 1 000 : ( 1 + 000) 1 000 2.718 x 0 (1 + x) 1 x a 18 / 27
70 x: 0 x 0 = (1 + x) 1 x e : lim x 0 (1 + x) 1 x = e () = 2.718281828459045... y = a x (0, 1) (0, 1) 1 a e 19 / 27
70 70 e () log e e rn ( log 1 + r ) r r 0 rn log e log 2 log e = 1 log 2 n log 2 r = log 2 20 / 27
70 log 2 : : : y O x 21 / 27
e x 70 n = 1, 2, 3... e x : e x 1 + x + x 2 2! + x 3 3! + + x n n! n! = 1 2 3 n (n ) x = 1 e n = 1: e 2, n = 2: e 2.5, n = 3: e 2.667 n = 4: e 2.708, n = 5: e 2.717, n = 6: e 2.718 n = 7: e 2.718 n 22 / 27
log 2 1 70 n = 1, 2, 3... log(1 + x) : log(1 + x) x x 2 2 + x 3 1 < x 1 x = 1 log 2 3 + x n ( 1)n+1 n n = 1: log 2 1, n = 5: log 2 0.5 n = 10: log 2 0.646, n = : log 2 0.688 n = 200: log 2 0.691, n = 300: log 2 0.691 n = 400: log 2 0.692, n = 500: log 2 0.692 n = 600: log 2 0.692, n = 700: log 2 0.692 n = 800: log 2 0.693, n = 900: log 2 0.693 23 / 27
log 2 2 70 log 2 log 1 + x 1 x : log 1 + x (x 1 x 2 + x 3 3 + x 5 5 + + x 2n+1 ) 2n + 1 1 < x < 1 x = 1 3 log 2 n = 1: log 2 0.691, n = 2: log 2 0.693 24 / 27
70 70 X r % n 2 ( 1 + r ) n = 2, rn ( log 1 + r ) r = log 2 r 0 rn log 2 log 2 0.693 n 70 r 25 / 27
70 ( 1 + r ) n = 2 n r log 2 n = ( log 1 + r ) 1 r r log 2 C = ( log 1 + r ) r = 0.1: C 69.3, r = 1: C 69.7, r = 2: C 70.0 r = 3: C 70.3, r = 4: C 70.7, r = 5: C 71.0 r = 10: C 72.7, r = 20: C 76.0, r = 30: C 79.3 r 70 26 / 27
40 70 X r % 1.5 40 r 27 / 27