() 3 3 2 5 3 6 4 2 5 4 2 (; ) () 8 2

4 0 0 2

ex. 3 n n =, 2,, 20 : 3 2 : 9 3 : 27 4 : 8 5 : 243 6 : 729 7 : 287 8 : 656 9 : 9683 0 : 59049 : 7747 2 : 5344 3 : 594323 4 : 4782969 5 : 4348907 6 : 4304672 7 : 294063 8 : 387420489 9 : 6226467 20 : 348678440.. 3 (i),000 (ii),000,000 (iii),000,000,000 0. (i) 7 (ii) 3 (iii) 9 3

; (n + ) 0 0 0 3 n 3 6 = 729 < 000 < 3 7 = 287 7 729 287 (i) 7. 3 () 4

2. 2.. (i),000 7 (ii),000,000 3 (iii),000,000,000 9. (i) 3 x = 000 x x = 6.2877 6 6 54 8 (ii) (iii) 3 x = 000000, 3 x = 000000000 x x = 2.5754 x = 8.863 2 3 48 36 8 20 42 54. 3 x = 000 x = 6.2877 5

log 3 (000) = 6.2877 6.2877 6.2877 6 24 (/) 0.2877 () 24 (/) = 6.9048 6 6 6 60(/) 6.2877 =6 6 54 8 () 3 3 log 3 (000) = 6.2877 log 3 (000) = log 0(000) log 0 (3) log 3 (000) = ln(000) ln(3) ( ( = = ) 3 0.477 ) 6.9077.0986 6

[] (i) a p a q = a p+q (ii) (a p ) q = a pq [] (i) log a (P Q) = log a (P ) + log a (Q) (ii) log a (P q ) = q log a (P ) [] log a (b) = log c(b) log c (a) 5 ex. (i) 3 4 3 6 = 3 0 4 6 {}}{{}}{ 3 4 3 6 = (3 3 3 3) (3 3 3 3 3 3) = 3 4+6 ex.2 (ii) (3 4 ) 3 = 3 2 { }} { { 4 }} { { 4 }} { { 4 }} { (3 4 ) 3 = (3 3 3 3) (3 3 3 3) (3 3 3 3) = 3 4 3 3. 3 4 = 8 log 3 (8) = 4, 3 6 = 729 log 3 (729) = 6 ex.3 (i) log 3 (8 729) = log 3 (8) + log 3 (729) ex.4 (ii) log 3 (8 3 ) = 3 log 3 (8) (i) (i), (ii) (ii) 7

ex.5 9683 7747. 9683 = 3 9, 7747 = 3 9 + = 20. 3 20 = 348678440 9683 7747 = 348678440 6. 3 00 log 0 (3) = () p5. 2., 000, 000, 000, 000, 000, 000 6.2877, 2.5754, 8.863 7, 3, 9 3 3 2 8

2 log 3 (000) = 6.2877 log 3 (000000) = 2.5754 log 3 (000000000) = 8.863 (ii), 000, 000 =, 000 2,, 000, 000, 000 =, 000 3 (ii) log 3 (, 000, 000) = 2 log 3 (, 000), log 3 (, 000, 000, 000) = 3 log 3 (, 000) 2 2 3 3 [] c log a (b) = log(b) log(a) log a (b) = log(b) log(a) log c (a) log a (b) = log c (b) log c (a) log a (b) = log c (b) () 9

log a (a) = 5 P = a p log a (P ) = p Q = a q log a (Q) = q P, Q P Q = a p (i) q a = a p+q log a (P Q) = p + q (i) log a (P Q) = log a (P ) + log a (Q) = log a (P ) + log a (Q) P q P q = (a p (ii) q ) = a pq log a (P q ) = pq (ii) log a (P q ) = q log a (P ) = q log a (P ) p P = a p log a (P ) = p P = a log a (P ) ( P ) p = log a (a p ) (ii) 0

P b c b = a log a (b) log c (b) = log c ( a log a (b) ) (ii) = log a (b) log c (a) log c (a) log a (b) = log c (b) 5 [] (iii) ap a q = ap q (iv) a q = a q [] (v) a = a, a 0 = ( ) ( P (iii) log a = log Q a (P ) log a (Q) (iv) log a P (v) log a (a) =, log a () = 0 ) = log a (P ) [] log a (b) = log b (a) log b (a) log a (b) = (iv),(v) 4 a 0 =, log a () = 0, ( ) a = p a p, log a = log Q a (Q) 6 5 a q (i) p q a = a q+(p q) = a p a q (iii) ap a q = ap q

a p a q = ap a q (iii) = a p q (i) p+( q) = a = a p a q a p (iv) a q = a q. ex. 3 = 3 log 3 (3) = ex.2 ( ) 3 5 243 3 = 2 35 2 log 3 = log 9 3 (243) log 3 (9) 3 2 32 = (iii) 32 3 = 2 32 2 = 3 0 ex.3 3 0 = log 3 () = 0 () (vi) a m n = n a m = n a m () (vi) log b ( n a) = n log b(a) () 4. 20 a > x < x 2 a x < a x 2 2

(a = 3, 9 ) y 9 x 3 x 9 9 x = 3 2x x x A B O 2 x y = 9 x, y = 3 x y = 9 y = 9 x x = y = 3 x x = 2 9 = 3 2 9 x = (3 2 ) x = 3 2x y = 9 x A y = 3 x B y y B y A 2 () y = 9 x x 2 y = 3 x 3

y ( x ) b = f a y = f(x) x a y b y = 3 x = 9 x 2 y = 9 x x 2 y = 3 x 2 2 3. y = x y 3 y 3 = x ) 3 y = 3x 3 5 HP http://www.sit.ac.jp/user/konishi/jpn/l Support/L Support.html B3 4

5 0 20 30 40 50 60 70809000 x log(x) > 4. log(5) 5 0 20 30 40 50 60 70809000 5 0 log(0) ) (i) log(5) + log(0) = log(5 0) 5 0 = 50 5. 3 5 0 20 30 40 50 60 70809000 3 3 3 ) (ii) 4 log(3) = log(3 4 ) 3 4 = 8 5

y = 3 x, y = 9 x 00 80 70 60 50 40 30 20 (ii) y = 9 x (i) y = 3 x 0 8 7 6 5 4 3 2 0.8 0.7 0.6 0.5 0.4 0.3 0.2 5 4 3 2 0 2 3 4 5 0. y = 9 x 2 y = 3 x 6

y = x, y = x 2 y = x 3 3 y = x α ( α ) y = x 3 y = x 2 y = x 0 8 7 6 5 4 3 2 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 0. 0.2 0.3 0.4 0.6 0.8 2 3 4 5 6 7 890 7

; 3 n (n =, 2,, 00) : 3 2 : 9 3 : 27 4 : 8 5 : 243 6 : 729 7 : 287 8 : 656 9 : 9683 0 : 59049 : 7747 2 : 5344 3 : 594323 4 : 4782969 5 : 4348907 6 : 4304672 7 : 294063 8 : 387420489 9 : 6226467 20 : 348678440 2 : 0460353203 22 : 338059609 23 : 944378827 24 : 28242953648 25 : 847288609443 26 : 254865828329 27 : 7625597484987 28 : 2287679245496 29 : 68630377364883 30 : 2058932094649 3 : 67673396283947 32 : 853020888584 33 : 5559060566555523 34 : 66778699666569 35 : 5003545098999707 36 : 500946352969992 37 : 450283905890997363 38 : 3508577672992089 39 : 40525555308976267 40 : 25766545905692880 4 : 3647299637770786403 42 : 0948989352359209 43 : 328256967394537077627 44 : 98477090283623288 45 : 295432706550833698643 46 : 8862938965250095929 47 : 2658884358957503287787 48 : 7976644307687250986336 49 : 23929932923067529590083 50 : 7789798769852588770249 5 : 25369396307555776630747 52 : 6460888922667329893224 53 : 938324566768009896796723 54 : 584973700304005969039069 55 : 74449200920790770507 56 : 52334763302736053723552 57 : 570042899082086640534563 58 : 4702869724624483492603689 59 : 430386097387345047648067 60 : 423958275262035429443320 6 : 277347482564860542883299603 62 : 3852042447694583628649898809 63 : 4456273430837494885949696427 64 : 34336838202925248465784908928 65 : 03005460877537453973547267843 66 : 3090354382632623692064803529 67 : 92709463478978370857692540587 68 : 2782838944369352572857762376 69 : 834385683308053377857328695283 70 : 25035550499324603557986085849 7 : 75094665497972480394675958257547 72 : 225283995449397448404787477264 73 : 67585986348752323552044362437923 74 : 20275559590445256970656330872953769 75 : 6082667877335770996839926886307 76 : 82480036340073273590597785658392 77 : 547440089420293820775593356975763 78 : 64232032682606584623467800709255289 79 : 492696098047897443869440340227765867 80 : 4780882944345923360832020638329760 8 : 4434264882430377699482496306949892803 82 : 330279464729330984474889857449678409 83 : 399083839487339929534246675572349035227 84 : 972558256209788602740026770470568 85 : 35975455476860593658082200805437043 86 : 07752636643058780974246602404534239529 87 : 323257909929745342922739807236027853387 88 : 9697737297875236028768294264080855606 89 : 29093289362570808630465826492242446680483 90 : 8727963568087724258939747947672734004449 9 : 268389070426337277674924384308202024347 92 : 7855672278948330225773529054606037304 93 : 2356550633836823549906773945876388923 94 : 7069650490504706497203958376494543357369 95 : 22089547045349496095875284474363007207 96 : 636268544359423584748287625385342308902632 97 : 908805632340782707542448628765602692670648963 98 : 5726468970223482262734588628468080780946889 99 : 779250690670443678820376588540424234035840667 00 : 553775207320330364629765622727020752200 9 4 log 0 (3) = 0.477 0.477 2 = 0.9542 5 8

( ) AU (AU) 00 80 70 60 50 40 30 20 0 8 7 6 5 4 3 2 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0. 9

6 (i) a p a q = a p+q (ii) (a p ) q = a pq (iii) ap a q = ap q (iv) a p = a p (v) a = a, a 0 = (vi) a m n = n a m = n a m (i) log a (P Q) = log a (P ) + log a (Q) (ii) log a (P q ) = q log a (P ) log a (b) = log c(b) log c (a) ( ) ( ) P (iii) log a = log Q a (P ) log a (Q) (iv) log a = log P a (P ) (v) log a (a) =, log a () = 0 (vi) log b ( n a) = n log b(a) log a (b) = log b (a) log b (a) log a (b) = y ( x ) b = f a y = f(x) x a y b y b = f (x a) y = f(x) x +a y +b 20