8 8 3 1 1.1 l x a b l = ax + b l x x l a b l = ax + b 5 cm 0 10 1 0.5 cm 1 x l l = 0.5x 5cm 5 5 0.5cm 1 x l l = 0.5x 1.5 値 x 値 0 4 6 8 10 値 x 値 5 10 15 0 5 長さ l 長さ0 1 3 4 5 cm 長さ l 長さ0 1 3 4 5 cm 1 1 1
5cm 1 100 3 5/99 cm 1 x l l = 5/99x 5/99 値 x 値 0 40 60 80 100 長さ l 長さ0 1 3 4 5 cm 3 3 x l a b c x y y = a log c x + b log c 10 x l a b l = a x + b (1) x l x l 1 0 3 5cm 1 100 4 x l l =.5 x 1 1 = 10 0.5 1 = 0 10 0.5.5 0.301 = 0.755 4 0.75 cm 11 1 1 10 11.5 11.5 1.041 =.605 値 x 値 x 値 11 10 0 3 45 10 1 0 10 0 1 長さ l 長さ0 1 3 4 5 cm 4 1
10.5 cm 4 0.1cm 1 10 10 0 5cm 5-5 5 x l l = (5/ 5) x 5 値 x 値 5 6 7 8 910 0 x 値 0.7 0.8 0.9 1.0 1.1 1. 1.3 1.4 長さ l 長さ0 1 3 4 5 cm 5.1 (1) x 6 x y y = ax + b y = ax b y = ab x y = a log e x + b 1 y = 90x + 500 () y = 0.8x (3) y = 0.9 1.1 x (4) 1 log a a y = a log e x + b = a log e 10 x + b = a (.30 ) x + b = a x + b A. 3
10000 4 10 4 y (linear) 5000 Y (= y) 3 1 y (logarithmic) 10 3 10 10 1 0 0 10 0 0 50 100 x (linear) 0 50 100 x (linear) (a) (b) 10000 4 10 4 y (linear) 5000 Y (= y) 3 1 y (logarithmic) 10 3 10 10 1 0 0 10 0 10 0 10 1 10 x (logarithmic) 10 0 10 1 10 x (logarithmic) 0 1 X (= x) 0 1 X (= x) (c) (d) 6 y = 000 log e x + 1000 (5) x y 10 X Y X = x y x = 10 X 4
y = 10 Y X Y x y.1.1-7 y = 0 log x 10000 y = 90 x + 500 y = 0.8 x y = 0.9 1.1 x y (linear) 5000 0 y = 000 log e x + 1000 0 50 100 x (linear) 7.1. - y 8 () y = 10 Y y = 90x + 500 10 Y = 90x + 500 10 (90x + 500) 90 + (x + 500/90) Y 1.954 + (x + 5.556) Y x 8-7 5
4 10 4 Y (= y) 3 1 0 y (logarithmic) 10 3 10 10 1 10 0 y = 90 x + 500 y = 0.8 x y = 0.9 1.1 x y = 000 log e x + 1000 0 50 100 x (linear) 8 (3) y = 10 Y y = 0.8x 10 Y = 0.8x Y = x + 0.8 Y x 0.09691 Y x 8-7 (4) y = 10 Y y = 0.9 1.1 x 10 Y = 0.9 1.1 x Y = x 1.1 + 0.9 Y 4.139 10 x 0.04575 Y x 8 10 Y 1 x 100 Y (= y) 4.139 x 100 Y (= y) 4.139 a b y = ab x 10 Y = ab x a + x b 6
b a a b (5) y = 10 Y y = 000 log e x + 1000 10 Y = 000 log e x + 1000 Y = 3 + (log e x + 1/) Y 0.9031 + (log e x + 0.5000) 8.1.3 - x 9 10000 y = 90 x + 500 y = 0.8 x y = 0.9 1.1 x y = 000 log e x + 1000 y (linear) 5000 0 10 0 10 1 10 x (logarithmic) 0 1 X (= x) 9 () x = 10 X y = 90x + 500 y = 90 10 X + 500 y 10 X+1.954 + 500 y X 9-7 7
(3) x = 10 X y = 0.8x y = 0.8(10 X ) y = 0.8(10 ) X y = 0.8 100 X y X 9-7 (4) x = 10 X y = 0.9 1.1 x y = 0.9 1.1 10X y = 0.9 (1.1 10 ) X y 0.9.594 X y X 9-7 (5) x = 10 X y = 000 log e x + 1000 y = 000 log e 10 X + 1000 y = 000X log e 10 + 1000 y 000.303X + 1000 y 4606X + 1000 y X 9 a b y = a log e x + b y = a log e 10 X + b y = (log e 10)aX + b (log e 10)a b a b 8
.1.4 - x y 4 10 4 y = 90 x + 500 y = 0.8 x Y (= y) 3 1 y (logarithmic) 10 3 10 10 1 0 10 0 y = 0.9 1.1 x y = 000 log e x + 1000 10 0 10 1 10 x (logarithmic) 0 1 X (= x) 10 () x = 10 X y = 10 Y y = 100x y = 90x + 500 10 Y = 90 10 X + 500 10 (90 10 X + 500) (90 10 X + 500) 10 (3) x = 10 X y = 10 Y y = 0.8x 10 Y = 0.8 (10 X ) 10 {0.8 (10 X ) } 0.8 + (10 X ) Y X 0.09691 Y X a b 9
y = ax b 10 Y = a(10 X ) b 10 Y = a(10 X ) b a + 10 b X a + bx b a a b (4) x = 10 X y = 10 Y y = 0.9 1.1 x 10 Y = 0.9 1.1 10X (0.9 1.1 10X ) 0.9 + 10 X 1.1 Y 0.04139 10 X 0.04576 Y X - 7 (5) x = 10 X y = 10 Y y = 000 log e x y = 000 log e x + 1000 10 Y = 000 log e 10 X + 1000 10 Y = 000X log e 10 + 1000 (000X log e 10 + 1000) Y (4605X + 1000) Y (X + 0.17) + 0.3663 Y X - 7. 1 y = ax + b 7 10
8 y = ab x 9 y = a log e x + b 10 y = ax b 1 X Y X 1 Y X = 0 Y -0.1-0.1 1 b = a = 0.1 a = 10 0.1 0.794 y = 0.8x 1 - y = ax + b a b - y = ab x b a - y = a log e x + b (log e 10)a b - y = ax b b a.3.3.1 - - y = 0.9 1.1 x (4) x 10 (6) y 000 (7) (8) (9) y = 0.9 1.1 (x 10) (6) y = 0.9 1.1 x + 000 (7) y = 3 1.1 x (8) y = 0.9 x (9) 11
(6) y = 10 Y 10 y = 0.9 1.1 (x 10) 10 Y = 0.9 1.1 (x 10) 0.9 + 1.1 (x 10) 0.9 + (x 10) 1.1 Y = x 1.1 + 0.9 10 1.1 Y 4.139 10 x 0.4597 11(b) 1 (7) y = 10 Y 10 y = 0.9 1.1 x + 000 10 Y = 0.9 1.1 x + 000 Y = x (0.9 1.1 x + 000) 11(b) x x 0 (8) y = 10 Y 10 y = 3 1.1 x 10 Y = 3 1.1 x (3 1.1 x ) 3 + x 1.1 Y 4.139 10 x + 0.4771 1 11(b) (9) y = 10 Y 10 y = 0.9 x 10 Y = 0.9 x (0.9 x ) y = 0.9 1.1 x 10 = 0.9 1.1 x 1.1 ( 10) 0.3470 1.1 x x 1
0.9 + x Y 0.3010x 4.576 10 1 11(b) x y Y axis (linear) 15000 10000 5000 y = 0.9 1.1 x y = 0.9 1.1 x -10 y = 0.9 1.1 x +000 y = 3 1.1 x y = 0.9 x Y axis (logarithmic) 10 10 10 5 y = 0.9 1.1 x y = 0.9 1.1 x -10 y = 0.9 1.1 x +000 y = 3 1.1 x y = 0.9 x 0 10 0 0 50 100 X axis (linear) 0 50 100 X axis (linear) (a) (b) 11 -.3. - - y = 0.8x (3) x 10 (10) y 0 (11) (1) (13) y = 0.8(x 10) (10) y = 0.8x + 000 (11) y = 1.3x (1) y = 0.8x 3 (13) (10) x = 10 X y = 10 Y y = 0.8(x 10) 13
10 Y = 0.8(10 X 10) 10 {0.8(10 X 10) } Y = (10 X 10) + 0.8 Y X 1(b) X 3 x X X (11) x = 10 X y = 10 Y y = 0.8x + 000 10 Y = 0.8(10 X ) + 000 10 {0.8(10 X ) + 000} {0.8(10 X ) + 000} Y X 1(b) (10) X 3 y X X (1) x = 10 X y = 10 Y y = 1.3x 10 Y = 1.3(10 X ) 10 {1.3(10 X ) } Y = X + 1.3 Y X + 0.1139 Y X 1(b) 1 x 1(b) (13) x = 10 X y = 10 Y y = 0.8x 3 10 Y = 0.8(10 X ) 3 10 {0.8(10 X ) 3 } Y = 3X + 0.8 Y 3X 0.09691 14
Y X 1(b) 1 x 1(b) x y Y axis (linear) 15000 10000 5000 y = 0.8x y = 0.8(x - 10) y = 0.8x + 000 y = 1.3x y = 0.8x 3 Y axis (logarithmic) 10 6 10 4 10 y = 0.8x y = 0.8(x - 10) y = 0.8x + 000 y = 1.3x y = 0.8x 3 0 10 0 0 50 100 X axis (linear) 10 0 10 1 10 X axis (logarithmic) (a) (b) 1 - A A.1 log log log b c c b c = b a c b log b c = log b b a = a a 10 100 = 10 = 78 10.444 =.444 10 = 10 1 = 1 1000 = 10 3 = 3 15
0.1 = 10 1 = 1 1 = 10 0 = 0 log log 8 = log 3 = 3 log 64 = log 6 = 6 log 1 = log 0 = 0 1 log 0.5 = log = log 1 = 1 1 log 0.5 = log 4 = log 1 = log = A. log A..1 log log a b c = c log a b b = a n n log a b = log a a n = n log a b c = log a (a n ) c = log a a nc = nc log a b = n = c log a b log 8 = log 3 = 3 log 16
A.. log log a bc = log a b + log a c a B = b a C = c A B bc = a B a C bc = a B+C a log a bc = log a a B+C log a bc = B + C B = log a b C = log a c log a bc = log a b + log a c A..3 a b log b a d = a h (14) a log log a d = log a a h log a d = h (14) d = a log a d b log log b d = log b a log a d log b d = log a d log b a log b a a b A.3 e e a e d dx ax = a x (15) 17
(15) d a x+h a x a h 1 = lim = a dx ax x lim h 0 h h 0 h (15) (16) (16) a lim k 0 a h 1 lim = 1 h 0 h (16) h lim h 0 a h 1 = 1 h = log a(1 + k) h 0 k 0 log a h = 1 + k log a (1 + k) = 1 k lim log a (1 + k) k 1 = 1 k 0 lim(1 + k) k 1 = a k 0 (17) (17) a e e e = lim(1 + k) k 1 k 0 k = 1/m m ( e = lim 1 + 1 m (18) m m) m (1 + 1/m) m 13 m.7 3.0.5 (1 + 1 / m ) m.0 1.5 1.0 10-3 10-10 -1 10 0 10 1 10 10 3 m 13 e (18) a n = (1 + 1/n) n (18) a n = ( a + 1 n) n 18
= n ( ) 1 nc p p=0 p=0 n p n ( ) n! 1 = p!(n p)! n p = 1 + n 1 1! n + n(n 1)! C n! (n p)! 1 n(n 1) {n (p 1)} 1 n(n 1) 1 1 + + + + p n p! n n! = 1 + 1 n 1! n + 1 n(n 1) + + 1 n(n 1) {n (p 1)} + + 1 n(n 1) 1! n p! n p n! n n = 1 + 1 n 0 + 1 1! n! = 1 + 1 1! ( 1 0 n n 0 n 1 + + 1 n n p! + 1 n (n 1) n (n ) n (n 1) n! n n n n ) + 1 ( 1 0 ) ( 1 1 ) + + 1! n n p! + 1 n! n 0 n 1 n (p 1) + n n n ( 1 0 ) ( 1 1 ) ( 1 n ) ( 1 n 1 ) n n n n ( 1 0 ) ( 1 1 ) ( 1 p 1 n n n n n ) + (19) (19) a n n a n a n (19) 1/1! 1/! 1/p! 1/n! (1 0/n) (1 0/n)(1 1/n) (1 0/n)(1 1/n) {1 (p 1)/n} (1 0/n)(1 1/n) {1 (n )/n}{1 (n 1)/n} {1 (p 1)/n} 1 1 1 1 1/1! 1/! 1/p! 1/n! a n 1/1! 1/! 1/p! 1/n! a n 1 + 1 1! + 1! + + 1 n! (0) 3! = 3 1 = 3 > = 4! = 4 3 1 = 4 3 > = 3 p! = p(p 1)(p ) 3 > p 1 (0) a n 1 + 1 1 + 1 + + 1 ( ) n 1 n 1 1 = 1 + 1 1 { ( ) n } 1 = 1 + 1 = 3 1 n 1 < 3 a n a n (18) e 19
A.4 + = = ( = 1 + ) 100 ( = 1 + ) 100 = = 10 000 1% 0 000 n 10000 1.01 n = 0000 1.01 n = 1.01 log 1.01 1.01 n = log 1.01 n 69.66 log 1.01 70.1. [1] 0
[1],,, 006. - - 4.0 http://creativecommons.org/licenses/by-nc-nd/4.0/deed.ja http://camellia.thyme.jp 1