#수Ⅱ지도서-4단( )

IV 4

3 4 5

5 exponent 3 3 Archimedes B.C. 87~B.C. Diophantos?00~?84 a m _a n =a m+n (mn=0y) Stifel M. 487~567 Arithmetica integra y-3--03y y ;8!; ;4!; ;!; 48y Stevin S. 548~60 xx x ()()(3) x ;!; x ;3!; x ;3@; {;!;}{;3!;}{;3@;} a a n a - a - ya ;!; a ;3@; y NewtonI. 64~ 77 (DescartesR. 596~650) Kepler J. 57~630 prosthaphraeresis Mirifici logarithmorum canonis descriptio Logarithm Log 858 TodhunterI. 80~884 (NapierJ. 550~67) log logos arithmos logarithm 544 Arithmetica integra 3 4 5 6 7 8 9 0 4 8 6 3 64 8 56 5 04 BriggsH. 56~630 0 64 0000 90000 00000 4 Arithmetica logarithmica 6

HEves 6 «'a logån log N 7

7 PPT PPT 8 8 x =8 n «'a 8

0 9 7.35_0 kg 0 4 0 36 0 8 0 40 0 0 44 0 6 0 48 0 0 0 5 0 4 0 56 0 8 0 60 0 3 0 64 9

_ 3 = +3 = 5 ( ) 3 = _3 = 6 3 = = 3-6 4 3 4 6 4 6 = ={ } 4 = 4 3 3 4 5 6 ;!; 4 4 x =4 x= x=- 4-4 '4 '4= - '_'8=' 6=4 "ç _3="ç _'3='3 '8 8 =æ ='4= ' 3 '3 '3 æ = = "ç 4 '3 '3 0

a m a n a+0 aa 8 3 =8 8 x 3 =8 (x-)(x +x+4)=0 x= x=- '3i - '3i a b 3 _ab =a _b 3 _a_b =a + _b 3+ =a 3 b 5 (a b) 3 =(a ) 3 b 3 =a 6 b 3 a 3 b (ab) =a 3 b a b =a 3- a _ = b b b 6 a 3 b - a 3 b 6 b 3 b 6 ab 3 { } 3 =ab 3 =ab 3 _ =a +3 _ = a4 a x+0y+0 b 3 b+0 a+0 -x 4 [{ } 3 ] 0x _x 6 (-x 5 ) y 64x 4 5 y 6 x

x =a x a x n =a x a n 3 a n n a -b =(a+b)(a-b) a 3 +b 3 =(a+b)(a -ab+b ) a 3 -b 3 =(a-b)(a +ab+b ) a 4 -b 4 =(a +b )(a -b ) =(a +b )(a+b)(a-b) 3 a n a n 9 x =9 x -9=0 (x+3)(x-3)=0 x=-3 x=3 9-33 -7 x =-7 x +7=0 (x+3)(x -3x+9)=0 3 3'3i x=-3 x= 3 3'3i -7-3 8 x =8 x -8=0 (x+3)(x-3)(x +9)=0 4 x=-3 x=3 x=-3i x=3i 8-33-3i3i a n 5 4 y=x n n n (-x) n =-x n y=x n n (-x) n =x n y=x n y y y O x O x f(x)=g(x) HjK y=f(x)y=g(x) x

5 6 7 a nn a a n HjK x n =a n a HjK «'a n a n= a 'a 'a a>0 n «'a a<0 n «'a n a «'a x n =a n a>0 «'a -«'a n «'a 7 «'a x 5 =3 x= fi ' 3= x 3 =0.008 x=0. 'ƒ0.008=0. x 3 =-8 x=- ' -8=- x 4 =0.000-0.0. 6 '4 - 'ƒ0.000=-0. ''8 '8 ' n -7-3 -8 8 '8 6 "ç(-) 3 =- "ç 3 "ç(-3) 4 =fi"ç(-3) 5 3

8 9 a>0 «'a («'a) n =a «'a>0 8 a>0b>0 n 'a'b=' ab 'a a =Æ 'b b ('a) n ="ça n 9 0 n a a>0b>0 n «'a { } n («'a)«a = = «'b («'b)«b «'a «'a>0«'b>0 >0 «'b «'a a n «'b b «'a «'b =«Æ a>0 m n {(«'a) m } n =(«'a) mn =(«'a) nm ={(«'a) n } m =a m («'a) m >0 («'a) m a m n («'a) m =«"ça m a b a>0 m n (μ "ç«'a) mn ={(μ "ç«'a) m } n =(«'a) n =aμ "ç«'a>0μ μ "ç«'a amn μ μ "ç«'a=μ «'a 0 a>0 b>0 «'a «'b=«' ab a=-9b=-4n= ' -9 ' -4=3i i=-6 " (-9)(-4)=' 36=6 ' -9 ' -4+" (-9)(-4) ' øπ'8= ' '8= ' 6= "ç = ( '9) "ç9 8 = = æ = ' 7= "ç3 =3 '3 '3 3 4

x x =6 x= ' 6= "ç = " _= ' x =( ') = "ç =4 '4-5 x =-5 x +5=0 (x+5)(x -5x+5)=0 5 5'3i x=-5 x= -5-5 ' 6= "ç = 5 5'3i ' ' 8= " _3 " _3 = " ( _3)_(_3 )= " _3 = " (_3) =6 '3 '3 = = Æ ;4 8; = Æ ; 6; = æ {;!;} 4 =;!; øπ' 48 ' 48 ' 6+ '= "ç + '= " _+ ' = '+ '=3 ' ' ' + ' 8= ' 4+ ' 8 = " _3+ " 3 _3 = '3+3 '3 =5 '3 ( '9+ '4)( '9- '4)=( '9) -( '4) ='9-'4=3-= ( '+ '3)( '4- '6+ '9) =( '+ '3)( "ç - 'ƒ_3+ "ç3 ) =( '+ '3){( ') - ' '3+( '3) } =( ') +( '3) =+3=5 n n n æ =a æ =b ab 3 n =a n 3 =b n=a =3b a 3 3b a 3 _3 =8_6 =7_9 =6_ =88_8 =648y 3b b 3_ =4 3_4 =9 3_6 =648y n 648 5

0 mm= m km=0 3 m 0 3 nm= m m=0 nm km=0 3 m 0 9 km=0 _0 nm=0 nm 0 0 0 0 0 ('3) =3 {;3@;} 0 = (-) -3 = =- (-) 3 8 { } - = = ;!; 6

3 a m a n =a m+n mn=0 m>0n<0 m<0n>0 m<0n<0 3 4 a m _a n =a mn a m a n =a m n 4 0 >0 x =- - 'ƒ- 'ƒ-=(-) ;!; -=('ƒ- ) ={(-) ;!; } ={(-) } ;!; = ;!; = -= +=0 a+0 m n p=-m q=-n p q a m a n =a -p a -q a m a n = a p a m a n = _aœ =a π_aœ a p a q a m a n =a -p+q =a m-n a m a n =a m-n m n a+0 b+0 n p=-n p (ab) n =(ab) -p = (ab) p = = a p b p a p b p =a -p b -p =a n b n (ab) n =a n b n 7

_4-6 = 4 _( ) - (_3) = 4 _ - ( _3 ) = 4-- 3 - = 0 3 - = = 3 9 (3 _9 - ) ={3 3 _(3 ) - } - =(3 3 _3-4 ) - =(3 - ) - =3 5 (a+a - ) =a +a a - +(a - ) =a ++a - (a+a - ) =a 3 +3a a - +3a (a - ) +(a - ) 3 =a 3 +3a a - +3a a - +(a - ) 3 =a 3 +3a+3a - +a -3 6 5 (a m ) n =a mn 6 a n =«"ça m a<0 n «'a a>0 (-) ;!; (-3) -;4!; m a n =«"ça m m n a -;#; a a -;#; =a -3 m -3 a -;#; =a -3 ="ça -3 (a>0) = - "ça 3 (a>0) ;4#; = "ç 3 3 0.5 =3 ;!; ='3 5 -;#; ="ç5-3 =æ = 5 3 "ç5 3 7 -. =7 -;5^; =fi "ç7-6 =fiæ = 7 6 fi"ç7 6 "ç 4 = ;3$; "ç3 4 =3 ;$; =3 "ç5-3 =5 -;4#; =fiæ =fi "ç7-3 =7 -;5#; fi"ç7 3 7 3 8

a>0 r s 7 m q r= s= (mqn p ) n p a r a s =a n a p = n "ça m p "ça q = np "çça mp np "ça nq = np "ç a mp a nq = np "ç a mp-nq =a =a - np =a n - p =a r-s a r a s =a r-s a>0 r s m q r= s= (mqn p ) n p (a r ) s =(a n ) p =( n "ça m ) = np "ça mq =a np =a n p =a rs (a r ) s =a rs m mp np m a>0 b>0 r m r= (mn ) n (ab) r =(ab) m n mq = n "ç(ab) m = n "ça m b m = n "ça mn "çb m =a n b n =a r b r (ab) r =a r b r q nq q m mp-nq np q q p m q m m 7 0 "ç _ ;3@; = ;3$; _ ;3@; = ;3$;+;3@; = ;3^; = =4 ( '3 ) 3 -;3!; =3 ;3@; 3 -;3!; =3 ;3@;-{-;3!} {(-3) }.5 =(-3) _.5 =(-3) 3 =-7 {(-3) }.5 =(3 ).5 =3 _.5 =3 3 =7 =3 ;3#; =3 =3 ;3!; _ '9_4 -;3!; =( _3) ;3!; _(3 ) ;3!; _( ) -;3!; = ;3@; _3 ;3!; _3 ;3@; _ -;3@; = ;3@;+{-;3@;} _3 ;3!;+;3@; = 0 _3 =3 ' 4 ;3!; _8 ;9!; = ;3!; ( ) ;3!; _( 3 ) ;9!; = ;3!; ;3@; _ ;3!; = ;3!;-;3@;+;3!; = 0 = 9

8 a>0b>0 mn «'a a «'a«'b=«' ab =«æ «'b b («'a) m =«"aω m μ "ç«'a =μ «'a 8 a ;!; (a -;!; ) =a ;!; a -;$; øπfl 'a_øπafi =Æ a ;6!; _a ;%; =a ;!;-{-;$;} =a ;%; =(a ;6!; _a ;%; ) ;!; =a ; ; _a ;4%; =a ; ;+;4%; =a ;3$; = "ça (a ;!; +a -;!; )(a ;!; -a -;!; )=(a ;!; ) -(a -;!; ) =a-a - (a ;3!; +a -;3!; )(a ;3@; -a ;3!; a -;3!; +a -;3@; )=(a ;3!; ) 3 +(a -;3!; ) 3 =a+a - 9 Æ '_ ' (5 ;4#; _0 ;!; 00 ;4!; ) ; ; 5'5 9 real number system R Q axiom of completeness r<r r r r r yr n y n r n r lim r n =r n a r«a r lim a r«=a r n n 30

0 a>0 (3 -;!; _4 '3 9 ;4!; _ -'3 ) '3 '3 3 4'3 3 a - _a - a -'3 8 3 -'3 _4 - =_( ) - =_ -4 = -4 = -3 =;8!; 3 (3-3 ) - =3 3 3 =3-3 =3 - =;3!; 0 a n 4 0.5 _8 ;!; =( ) ;4!; _( 3 ) ;!; = ;!; _ ;#; = ;!;+;#; = =4 ( ' ) '8 = ''8 = ' 6 = 4 =6 ' _4 ' = ' _( ) ' = ' _ ' = '+' = 3' (') 3' ' =( ;!; ) 3' ' = 3' -' = (3 '+ ) '- =3 ('+)('-) =3 - =3 ( ' 3 '6 ) ' _9 '3 = 3 '3 _(3 ) '3 ' = 3 -'3+'3 = 3 0 =4 a>0 {(-) } ;#; =4 ;#; =( ) ;#; = ;#; = 3 3

. a>0 a=0 a<0 n «'a 0 «'a n «'a-«'a 0. «"ça m x+yx-yxyxx a n 0 64 ' 64 a= ' 64=4 8 ' 8 b= ' 8=3 a+b=7 0 {-;!;} -3 =(-) 3 =- 3 =-8 4_(-) = 3 _ - = 3-+(-) = - =;!; 03 ' = æ ; 6; = æ;8!; =;!; ' 6 øπ '4 =" '8 '4=" ' 3 =fl ' 3=fl "ç 5 = ;6%; 04 05 øπ"ç'=æ ( ;!; ) ;!; =Æ +;!;+;4!; =( ;4&; ) ;!; = ;8&; øπ' '=( ;!; ;3!; ) ;! =( +;!;+;3!; ) ;!; =( : 6 : ) ;!; = ;!!; '=fl "ç =fl '8 '3=fl "ç3 =fl '9 fl'6<fl '8<fl '9 fl'6<'< '3 a +a - =(a+a - ) -=3 -=7 a 3 +a -3 =(a+a - ) 3-3 a a - (a+a - )=3 3-3 3=8 (a ;!; +a -;!; ) =a+a - +=5 a>0 a ;!; +a -;!; ='5 a ;#; +a -;#; =(a ;!; +a -;!; ) 3-3(a ;!; +a -;!; )=('5) 3-3'5='5 06 07 x a = 3 x =b yy` 7 x =( 3 ) x =( x ) (3 x a ) ={ } 3 b =;8!;a b 7 x =;8!;a b yy` 40 % 7 x a b 60 % a b c + + (a-b)(c-a) (b-c)(a-b) (c-a)(b-c) 3

08 (b-c)a +(c-a)b +(a-b)c = (a-b)(b-c)(c-a) (b-c)a -(b-c)(b+c)a+(b-c)bc = (a-b)(b-c)(c-a) (b-c)(a-b)(a-c) = =- (a-b)(b-c)(c-a) a a {x a-b } c-a _{x b-c } a-b _{x c-a } b a b c + + (a-b)(c-a) (b-c)(a-b) (c-a)(b-c) b c b-c =x =x - = x A=0.007_8 0.45 _9 0.75 =0.007_(3 4 ) ;4!0&; _( 6 _3) ;4@0(; =0.007_3 ;4^0*; _ ;*0&; _3 ;4@0(; =0.007_ ;*0&; _3 ;4(0&; a=;*0&;b=;4(0&; c k P=k Q " 8 Q=( ") k=( ; ; ) k= ;3@; k Q = ;3@; k = ;3@; P k " ( ") ; ; =( ; ; ) = 33

a _a 3 a=a 4 3a _a =36a 4 (x) 3 3y _4y= 3x 3 3y ab 0 xy ( a x =b y =6 { 3 + = 9 x y 4 ab {; 7;} ;#; -;9$; =;4(; ;!; 4 8 "Ω64 =6 "aω x a y x=+'3y=-'3 øπ 'a x 8 x =9 {; 6;} - ;3!; ;!; 3 'a 'a a a'a ab 'a=79 'b=5fl 'c=64 æ 3 c abc ;3!; ;!; 3 ab Á a Á b= b 'a abc (a Á b) Á c=a Á (b Á c) b 'a Á c=(a Á c) Á b c 'ƒa Á b=a Á (bc) ABC A='B= '3C=fl'6 A<B<C C<A<B A<C<B B<C<A C<B<A A={x x=56 n n 0 } 4 5 0 77 78 34

0 x =9{;5!;} y =7 xy øπ " _fl'8 {; 7;} -;3!; ;!; _{;3@;} ;!; 4 3 x + y 3 000 x +5 -y 0 x +5 -y a+a - = a>0 a ;!; +a -;!; a 3 +a -3 n f n (x) f n (x)=x n (f 3 Á f )(x)_( f 4 Á f 3 )(x)_y_( f 0 Á f 9 )(x) x x +px+6=0 a b p ab a ;5 0; x= + - ;3!; + -;3!; a>0'=.44 35

-0 x=+'3y=-'3 x+y=(+'3)+(-'3)=4 xy=(+'3)(-'3)=4-3= a x+y a "aω x a y = = =a øπ 'a a a xy x 3 +3x -: 4 :=0 a _a 3 a=a +3- =a 4 3a _a =(3_)_a + =6a 4 (x) 3 3y _4y=8x 3 3x 3 4y= 3y 3y {; 7;} ;#; -;9$; ={;3@;} 3_;#;_{-;9$;} ={;3@;} - =;4(; " 64 ={( 6 ) } ;3!; = 6 ;3!; = 4 =6 (a Á b) Á c= 'a Á c=ç " 'a= ç 'a a Á (b Á c)=a Á ç 'b= ç'b 'a + 'a Á c=ç " 'a= ç 'a(a Á c) Á b=ç 'a Á b= " ç'a= ç 'a = ç 'ƒa Á b=ç " 'a= ç 'aa Á (bc)= ç 'a = x =9 x 8 x {; 6;} - =( -4 ) - 8 = =( x ) ;!; =9 ;!; =3 x a ;3!; =3 6 b ;4!; = 9 c ;6!; = 6 56 n =( 8 ) n = n n=48 x= 4 8 x=4656 +4+6+56=78 8 ab 3 c æ =a ; 8; b ; 8; c -; 8; =(a ;3!; ) ;6!; (b ;4!; ) ;9@; (c ;6!; ) -;3!; =(3 6 ) ;6!; ( 9 ) ;9@; ( 6 ) -;3!; =3 - =3 A='B= '3C=fl '6 A= ;!; =( 3 ) ;6!; =8 ;6!; B=3 ;3!; =(3 ) ;6!; =9 ;6!; C=6 ;6!; C<A<B øπ " _fl '8=(_ ;4@; ) ;3!; _( 3 ) ;6!; =( ;#; ) ;3!; _ ;!; = ;!;+;!; = {; 7;} -;3!; ;!; _{;3@;} ;!; ={; 7;} -;6!; _{;3@;} ;!; = {;3@;} 3 -;6!; _{;3@;} ;!; ={;3@;} -;!; _{;3@;} ;!; ={;3@;} -;!;+;!; ={;3@;} 0 = ( a x =b y =6 yy { 3 + = 9 x y 4 yy a=6 x b=6 y a =6 x { a ;!; +a -;!; } =a+a - + a ;!; a -;!; =+=4 a b=6 x _6 y =6 x + y =6 ;4#; =( 4 ) ;4#; = 3 a>0 a ;!; >0a -;!; >0 ab a b= 3 = a=b= ab=4 a ;!; +a -;!; >0 a ;!; +a -;!; ='4= a 3 +a -3 =(a+a - ) 3-3a a - (a+a - ) = 3-3 =8-6= 36

x +px+6=0 6 a _ b =6 a+b = 4 a+b=4 ab a=b=3 a=3b= p=-( a + b )=-(+ 3 )=-0 3 (f n+ Á f n )(x)=f n+ (f n (x))=f n+ (x ) =(x n ) n+ =x n(n+) yy` (f 3 Á f )(x)_( f 4 Á f 3 )(x)_y_( f 0 Á f 9 )(x) x= + - = ;3!; + -;3!; = ;3@; -+ -;3@; x+= ;3@; + -;3@; yy (x+) 3 =( ;3@; + -;3@; ) 3 = + - +3( ;3@; + -;3@; ) = + - +3(x+) x 3 +3x +3x+=4+;4!;+3x+3 ( ;3!; + -;3!; )( ;3@; -+ -;3@; ) ;3!; + -;3!; =x 3 _x 3 4 _y_x 9 0 =x + +y+ 3 3 4 9 0 =x {;!;-;3!;}+{;3!;-;4!;}+y+{;9!;-; 0;} =x ;!;-; 0; =x ;5@; yy` x ;5@; x p 5 p x x= 5 =3 yy` (f n+ Á f n )(x) 4 5 x 4 x 3 +3x -: 4 :=0 m 0 tm t m t =m 0 a -t (a>0) yy` 0 x =9{;5!;} y =7 9 x =07 y =;5!; 3 3 x =03 y =;5!; yy` ;!;m 0 =m 0 a - a - =;!; t ;5 0; yy` 4 3 4 3 3 + =3 3 =(3 ) x y x y x 3 y =0 ;5!;=0 yy` 3 ;[@; 3 ;]#; 4 3 ;[$;+;]#; 3 000 x +5 -y 0 3x +5-3y = 0 x +5 -y 0 x +5 -y (0 x ) +(5 -y ) = 0 x +5 -y (0 x +5 -y )(0 x -0 x 5 -y +5 -y ) = 0 x +5 -y =0 x -0 x 5 -y +5 -y yy` 3 =9-9 7+7 =567 yy` m 0 a -t ;5 0;m 0 a -t ;5 0;(a - ) t ;5 0; {;!;} ;5 0; yy` t={;!;} = = >;5 0; 3' 45.48 t={;!;} 6 =;6 4;<;5 0; yy` m t 3 a - 4 t ;5 0; 4 4 t 37

= 0 0 9={;3!;} - - 0=(0 - ) - =(0.) - - (a ;!; ) ;3@; =a ;!;_;3@; =a ;3!; (a b ) ;!; =(a ) ;!; (b ) ;!; =ab a a a ;3!; ab '= ;!; =( ) ;4!; =4 ;4!; ;4!; 0 - - ;4!; a ;3!; _a ;3@; =a ;3!;+;3@; =a =a 'a_ 'a "ça 4 fl "ça "ç'a a ;%; a ;!; =a ;%;-;!; =a a ;6%; a a ;4!; 38

x =N x N x N 3 =8 9 =5 x 3 y 9 y g 4 8 y 5 y a x =N HjK x=log a N x a x log a N log a N a a>0a+ log =x x = x =x log 0 =x 0 x = 0 x =x log (-) =x (-) x = (-) x =x log a N N N>0 x=log (-4) x =-4 x x x >0 x =-4 x 39

3=log 5 0=log -=log 0.5 ;!;=log '3 =6 3 = {;3!;} - =9 9 ;!; =3 log 5=x 5 =5 5=5 x= log ;4 9;=x 7 =;4 9; ;4 9;=7 x=- x x= fi x=;3 ; x =7 x -7=0(x-3)(x +3x+9)=0 x x=3 3fl =79 5 =65 fi =;3 ; 6 ;!; ='6 6=log 79 4=log 65-5=log ;3 ; ;!;=log '6 x log x= log ;4!; x=-;%; 49 3 7 ph p h=6400_ln{ } 0 e ln h feet p pounds/feet 40

3 4 a>0 a+ M>0 N>0 log a M=plog a N=q a p =Ma q =N M a p = =a p-q N a q M log a =p-q=log a M-log a N N a>0 a+ M>0 log a M=p a p =M k (a p ) k =M k a kp =M k log a M k =kp=k log a M 4 log a MN+log a M log a N M log a M log a + N log a N log a M k +(log a M) k 3 a p =x log a x=p log a x a p p a p =xa q =y xy=a p a q =a p+q log a xy p+q a p a q pq log a xy=log a x+log a y log 5'5=log (5 _5 ;!; ) log º ' 0.000 =log 5 +;!; =log 5 ;%; =;%; log 5=;%; =log º " 0-4 =log º (0-4 ) -;!; =log º 0 = log º 0= 4

log 6+log ;!;=log {6_;!;} log 45-log 5=log : 5 : =log 3= =log 9 =log 3 = log ' 5-log 3=log ('ƒ5 ) -log 3 =log 5-log 3 =log : 3 : =log 5= log +log 6- log 3=log (_6)-log 3 _6 =log 3 =log =3 log =3 5 log 4=log =log =a log 0=log (_5) =log +log 5 =log + =a+ log =log ( _3) =log +log 3 = log +log 3 =a+b log ' 5=log 5 ;!; =;!; log (3_5) =;!; (log 3+log 5) =;!; (b+) 5 0 log c b log a b= p log c a log a b=p b=a p yy log c b =p log c b=p log c a log c a b=c plogça yy a c 4

log 3 log 4= log = log = log log log 4 log 5 log 6 log 4= log 6 = log 4 log 6 6 log 3 b log 3= = log a log '8=log ( ) ;!; =log ;#; =;#; log =;#; log log 3 = 3a b x= 3 =8 x 3 =5 x=5 6 a>0a+b>0c>0c+ log log a b= c b log c a log a b= b+ log b a n log aμ b«= log a bmnm+0 m a logåb =b a logçb =b logça a>0a+b>0b+c>0 log a b log b a= log a b log b c=log a c log 6+log ;3!;=log {6_;3!;} log -log 4=log : 4 : log 7=log 3 =3 log 3=3 log log aμ b n c b n nlog c b n = = = log a b log c a m mlog c a m n log aμ b n = log a b m a log c =x log c=logåx c log c x = logç x=log a log c b log c a x=c log a =log = =log 3= log '3 log 4=log 3 ;!; log =;!; log 3 log =;!; log 3 = log 3 a log c =c log a 43

0 000 μm M M=log º 000 =log º 0 =3 log º 0=3 log 0000=log 0 =4 log 0.000=log 0 =-4 log ' 00=log 0 ;3@; =;3@; log =log 0 -;3!; =-;3!; ' 0 Magnitude 3.7 Intensity Modified Mercalli MM 44

0.4969 0.649 0.7076 0.845 x=5.33 x=9.8? =.00 9.99 log 7.5=log (7.5_0)=log 0+log 7.5 =+0.8603=.8603 log 0.075=log (7.5_0 - )=log 0 +log 7.5 =-+0.8603=-.397 log 7500=log (7.5_0 )=log 0 +log 7.5 =4+0.8603=4.8603 log 0.00075=log (7.5_0 )=log 0 +log 7.5 =-4+0.8603=-3.397 45

log x=.4065=+0.4065=log 0+log.55 =log (.55_0)=log 5.5 x=5.5 log x=-.5935=-+(-0.5935)=-+0.4065 =log 0 +log.55=log (.55_0 - )=log 0.055 x=0.055 0 000_(+0.06) 0 log (000_.06 0 )=log 0 +0 log.06 =3+0_0.053 =3+0.506 =log 0 +log 3. =log (0 _3.)=log 30 30 log.=log (._0)=log 0+log. =+0.088=.088 log 000=log (._0 5 )=log 0fi +log. =5+0.088=5.088 log 0.0=log (._0 - )=log 0 +log. =(-)+0.088=-.97 log =log -log 0=-log 0 0 =-log (._0 )=-(log 0 +log.) =-(3+0.088)=-3.088 log N=n+a n 0 a< n a N N=log =0.300 0 log < n=0 N=0log 0=log (0_) =log 0+log =+0.300 log 0< n= N=00log 00=log (0 _) =log 0 +log =+0.300 log 00<3 n= N=000log 000=log (0 _) =log 0 +log =3+0.300 3 log 000<4 n=3 n=k (k ) N (k+) 46

. log a N log N M. 0 MN 3 k Na N 0 0 03 04 x->0x-+ <x<3 x>3 5-x>0 x<5 log x- (5-x) x <x<3 3<x<5 + =log +log 3=log (_3)= log 6 log 6 (log 8+logª 4) log 3=log 3 (log 3 +log 3 ) =log 3 (3 log +log ) =log 3 4 log =4 log 8370=log (83.7_0 )=+.97=3.97 log =log -log 83700=-log (83.7_0 ) 83700 =-(3+.97)=-4.97 log {-;!;}+log {-;3!;}+y+log {-;3 ;} 05 =log ;!;+log ;3@;+y+log ;3#!; =log {;!;_;3@;_y_;3#!;}=log ;3 ;=log fi =-5 x +4x+=0 log alog b log a+log b=-4log a log b= yy` log b log a (log a) +(log b) log a b+log b a= + = log a log b log a log b (log a+log b) - log a log b = log a log b (-4) - = =4 4 yy` 06 log a log b 40 % x =a logæa=logå x=;!; y =a logúa=3logå y=;3!; z =a logω a=4logå z=;4!; logå xyz=logå x+logå y+logå z=;!;+;3!;+;4!;=;!#; 60 % 47

a 0 % 50 (+0.) ;!0%0); a=. ;#; a 50 ;#; 07 08 log 8-log 3=log -logy yy log x-log 8=log 0-log yy _3 log y=log =log y= 8 0_8 log x=log =log 30 x=30 8 x=30y= A 50 m A(0.75) 0 log (0.75) 0 =0 log 0.75=0 log (7.5_0 - ) =0(-+0.875)=-0+8.75=-+0.75 =log 0 - +log 5.636=log (5.636_0 - ) =log 0.05636 (0.75) 0 =0.05636 5.636 % log. ;#; =;#; log.=;#;_0.079=0.88=log.35. ;#; =35.35 a n (+0.) 00 a=. 00 a n n n n. 00 n. 00 = n n log. 00 =log log.=log 00 n _0.079=0.300 n=380.050y 00 38 n 48

log x+ (3-x) x 3 4 5 ab log a+log b=8log a log b= log a +log b 3 4 6 8 0 log 0.009 log 4. log 6664 log 379000 log 0.0749 mn m+nmn 84 a< + <b a A m n b B log A+log B 5 6 7 8 9 A log A n a n a x -(log 8)x+k=0 3 n-a k : 8 : : 8ª: : 8 : x -4x+=0 ab log a+b {+a+ }+log a+b {+b+ } b a : 8 : : 8 : log '6 log 3 log 3 3 3 0 a 0 b 30 a+b log =0.300log 3=0.477 a 8% 0 0 log =0.30log.5=0.40log 3=0.48 8 9 0.5 3 4 5 49

log '6+;!; log ' 4-log 9 a log 8+log 5-log '3 log º a=b log º b ax -(3a-)x+4=0 a> log N log N a a N 4 C 4 4 C 5730 t 4 Cx% x=0 _ - t 5730 968 4 C% log =0.30log 3.3=0.5 5 a b +ab log a b+log b a= a +b b>a>0a+b+ ABC AB =4('3+) sin A=p sin B=q pq A B x log (k-) (kx +kx+) k log (+p)+log (4-4p)=0 q log q =4q 50

3 : : - 3 0 ;3*; a 0 x+>0x++ -<x<- x>- 3-x>0 x<3 log x+ (3-x) x -<x<- -<x<3 x0 0++=3 log 0.009=log (9_0-3 )=-3+log 9 log 4. log 6664=log (0 _6.664)=3+log 6.664 log 379000=log (0fl _.379)=6+log.379 log 0.0749=log (0 - _.749)=-+log.749 log 9 A log A n a n a x -(log 8)x+k=0 n+a=log 8 =log 3<log 8<log 9= n=a=log 8-3 n-a =3 -(log 8-) =3 3-log 8 =3 3 _3 log 8 =7_;8!;=: 8 : 3 log =0 log 3-30 log =0 0.477-30 0.300 =-4.59=-5+0.74 log 5=log : º:=-log =-0.300=0.6990 log 6=log +log 3=0.300+0.477=0.778 log 5<0.74<log 60.74=log 5. 3 log 5 5 a+b=5+5=0 log a +log b 3= + log a log b log a+log b 8 = = =4 log a log b 8 log (m+n)<94 log mn<5 0 m+n<0 0 mn<0fi m+n 0 m+n 0 + = < < m n mn 0fi mn 0 0 < + <0fi m n A=0 B=0fi log A+log B=3+5=8 x -4x+=0 a+b=4ab= log a+b {+a+ }+log a+b {+b+ } b a =log a+b {+a+ }{+b+ } b a ab+ ab+ =log a+b {+ }{+ } b a 4 =log {+ }{+ }=log {+ + + } b a a b ab (a+b) 4 =log [5+ ]=log {5+ }=log 3 ab a 0 a(+0.08) 0 log (.08) 0 08 =0 log.08=0 log =0 (log 08-log 00) 00 =0(log 08-)=0( log +3 log 3-) =0( 0.30+3 0.48-)=0.4 log.5=0.40 (.08) 0 =.5 0.5a.5 log '6+;!; log ' 4-log 9 = log 6 ;!; +;!; log ;!; -log 3 = ;!; log 6+;!; log -;@; log 3 ;!; =log 6+ log -log 3=log 6+log -log 3 6_4 =log =log 8=3 3 log 8+log 5-log '3=log +log 5 -log 3 ;!; =3+3-;!;=: : 5

log N=n+a n0 a< 3a- n+a= =3- =+{- } a a a - a 4 _{- }= a=3 a a log N=;3*; N=0 ;3*; log x=log (0 _ - t t 5730 )=- log yy` 5730 t=968 968 log x=- log =-.6_0.3=.5 yy` 5730 log 3.3=0.5 +0.5=log 0+log 3.3=log (0_3.3)=log 33. x=33. % yy` log a b=t b>a t> 5 log a b+log b a=;%; t+ = t t t -5t+=0 (t-)(t-)=0 t> t= log a b= a =b a b +ab a +b a b +ab b +ab = = = a a +b ab+b 4 t log x 4 log x x 5 log (k-) (kx +kx+) (k-) >0(k-) + k+k-+ k+k+3k+ yy kx +kx+>0 k>0 kx +kx+=0 D D=k -4k<0 k(k-4)<0 0<k<4 0<k<4 k=0kx +kx+>0 k0 log º b log º b< 0 b<0 yy yy` b=log º a 0 log º a<00 0 0 a<00 0 0 0 a<0 00 yy` a 0 00-0 0 yy` b 4 a 4 a 4 log (+p)+log (4-4p)=0 log (+p)(4-4p)=0 (+p)(4-4p)=4-4p =p =;4#; '3 A p= A=60 yy` q log q =4q log q log q =log 4q log q _log q=log +log q (log q) =+3 log q (log q) -3 log q-=0 ( log q+)(log q-)=0 log q=-;!; log q= q= -;!; q= ' B q= B=45 yy` A 6 B 9 5

3 ' 6_ 3 ' 54_ 3 Æ;4!;= ;3$; _(_3 3 ) ;3!; _{;!;} ;3@; = ;3$; _ ;3!; _3 ;3#; _ -;3@; = ;3$;+;3!;+{-;3@;} _3 =_3=6 log 6-log ;8#;=log (_3)-log ;8#; =log +log 3-(log 3-log 8) =log +log 8=log (_8) =log 6=4 øπ'a`øπa`3"ça 4 ={a ;!; (a a ;3$; ) ;!; } ;!; =(a ;!;+;!;+;3@; ) ;!; =a ;6%; k=;6%; a 3x -a -x a x a x -a -3x a x (a 3x -a -x ) (a x ) - 9- = = =3 a x (a x -a -3x ) a x -a -x 3-;3!; a=3 ;6!; b=7 ;5!; c= ;!; n (abc) n =(3 ;6!; _7 ;5!; _ ;!; ) n =3 6 _7 5 _ n n n n n n 3 6 _7 5 _ 6 5 n 65 30 n n x+y=4 y=4-x 3 x+ +9 y =3 x+ +3 y =3 x+ +3 4-x 3 x+ +3 4-x æ" 3 x+ 3 4-x = 3 3 =54 54 (log a+logªb) log ' ab 7 =log ab log ' ab 7 =log ab 3 ;!; log ab 3=6 53

O ABO AB =AO +BO ' 6 (å ) ={ } +(fl ' 08) =( ;3!; ) +{( _3 3 ) ;6!; } (å ) = ;3@; + ;3@; _3=4 ;3@; = +;3@; = ;3*; a = ;3*; a=;3*; a=;3$; a x =b y =c 5z = 0 xlog a=ylog b=5zlog c=0 0 5 =log a =log b =log c x y z 0 5 + + =log a+log b+log c=log abc=log 6=4 x y z log ab+log bc=log ab c=5 ab c=fi yy log bc+log ca=log abc =8 abc = yy log ca+log ab=log a bc=7 a bc= yy yy` a b c = abc=fi yy yy` b= c= =8 a= =4 a=4b=c=8 yy` 30 % abc 30 % abc 40 % 0 m T l=0t=;!;tº =;!; T 0 0=log a ;!; a 0 =;!; a={;!;} ; 0; = -; 0; ;4!;l l=log a ;4!; a= -; 0; l=log -; 0; - l= - log =0 -; 0; 0 m A T A B T B 30 m 30=log a -log a 30=log a a= -; 0 T A T B T A T 0 T B T 0 =( -; 0; ) 30 = -3 =;8!; T A :T B =:8 T A T B 54

4 a 0 % 0.a 5730 t 5730 0.a=a{;!;} ={;!;} 0 t 5730 t 5730 log =log {;!;} 0 t 5730 5730 -=- log t= = =900 5730 log 0.3 900 8000 55

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