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Page 1
Definition of rational indices
a
n
...n
a a a a a
1n
n
a
a
0a
1
nn
a a
0
1a 0a
( )
p
q qp pq
a a a p q 0q
1
33
(343...
Page 2
! ( ) 2x
f x
7
( ) ( )
3
x
g x ( ) 0.8x
h x
" 2x
y
" 1
( )
2
x
y
# # # $
%
1
8
1
4
1
2
& '
# # # $
% ' &
1
2
1
4
1
...
Page 3
" 2 , 3 4x x x
y y and y
" 1 1 1
( ) ,
2 3 4
x x
x
y y and y
" % ( ) * +
( ) 0.5x
f x
3
( ) ( )
4
x
g x ( ) 0.85x
h...
Page 4
( e
( ) x
f x e
1
lim(1 )x
x
e
x
2.718281828...e
x
y e x
y e
x
y e & x
y e
0 x
y e 1 x
y e
) x
y e x
y e % % 2. 3
)...
Page 5
( ) 10x
f x - (1)f ( 2)f
555555555555555555555555555555 555555555555555555555555555555
5555555555555555555555555555...
Page 6
Logarithm Function
f 6{( , ) / , 0 1}x
x y y a a and a f
1
f 1
f 6{( , ) / , 0 1}y
x y x a a and a /
7 - logay x y
...
Page 7
4. log log loga a a
x
x y
y
; 7
5. log n logn
a ax x : 7
log
6. log
log
b
a
b
x
x
a
< 7
1
7. log logn aa
x x
n
1
1
...
Page 8
: 5. log n logn
a ax x
7 loga x m
m
x a
( )n m n nm
x a a
log nmn
a x
( log n logn
a ax x
: log
6. log
log
b
a
b
x
...
Page 9
Example !
9
9
log 512
( )
1
log
32
a
2 5 7( ) log 125 log 49 log 16b
Solution
9
9 9
5
9
9
log 512 log 2
( )
1 log 2...
Page 10
( 10( ) logf x x $ < 7
( 10log x % % log x $
( ( ) logeg x x e ? 7
( loge x % % ln x
!
,@ =( %
" + "
10 2 1.3( ) l...
Page 11
( % . % % .
& / 1a ( ) logaf x x -
0 / 0 1a ( ) logaf x x " -
1 , R R
Example " ( ) x
f x e ( ) lng x x
"
( ( ) ln...
Page 12
2
1
log 8
256
10log 300 24771
!
3log 81 7
1
log
49
5
10log 10 loge e
& !
4 4
1
log 9 log
9
3 3log 108 log 4
log 25...
Page 13
$ ! 0x 0y
2
log3 log 4 log
x
x y
y
2
5 101
3ln 2ln ln
5
y
y x y
x
) 2
4 4 4
3
2log log log ,
2
x
x y x y
y
0x 0y y...
Page 14
3 3 3
4 2 2 3 2
2 5 23 3 2
(log 2 ) log 5 (log 3 ) log 2 log 3 log 2
2 1 2
(4) 2 3
3 3 3
8 2 1
5
Example - 10
1
2 ...
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Exponential and logarithm function

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Exponential and logarithm function

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Exponential and logarithm function

  1. 1. Page 1 Definition of rational indices a n ...n a a a a a 1n n a a 0a 1 nn a a 0 1a 0a ( ) p q qp pq a a a p q 0q 1 33 (343) 343 7 0 (999) 1 3 3 344 1 81 ( 81) 3 27 Laws of indices a b m n m n m n a a a m n m n a a a ( )m n mn a a ( )n n n ab a b ( ) n n n a a b b 0b Exponential Function 0a 1a ( ) x f x a x
  2. 2. Page 2 ! ( ) 2x f x 7 ( ) ( ) 3 x g x ( ) 0.8x h x " 2x y " 1 ( ) 2 x y # # # $ % 1 8 1 4 1 2 & ' # # # $ % ' & 1 2 1 4 1 8 x-4 -2 2 4 y -5 5 10 15 20 x-4 -2 2 4 y -5 5 10 15 20
  3. 3. Page 3 " 2 , 3 4x x x y y and y " 1 1 1 ( ) , 2 3 4 x x x y y and y " % ( ) * + ( ) 0.5x f x 3 ( ) ( ) 4 x g x ( ) 0.85x h x , - ( ) x f x a 0a 1a ( ) 0x f x a x ( % . (0,1) ( . % . & / 1a ( ) x f x a - 0 / 0 1a ( ) x f x a " - 1 , R R x-4 -2 2 4 y -5 5 10 15 20 x-4 -2 2 4 y -5 5 10 15 20
  4. 4. Page 4 ( e ( ) x f x e 1 lim(1 )x x e x 2.718281828...e x y e x y e x y e & x y e 0 x y e 1 x y e ) x y e x y e % % 2. 3 ) x y e x y e % % 4. 3 - ( )f x x ( ) 4x f x - (0)f (3)f 555555555555555555555555555555 555555555555555555555555555555 555555555555555555555555555555 555555555555555555555555555555 x-10 -5 5 10 y -10 -5 5 10 x-10 -5 5 10 y -10 -5 5 10 x-10 -5 5 10 y -10 -5 5 10 x-10 -5 5 10 y -10 -5 5 10 x-10 -5 5 10 y -10 -5 5 10 x-10 -5 5 10 y -10 -5 5 10
  5. 5. Page 5 ( ) 10x f x - (1)f ( 2)f 555555555555555555555555555555 555555555555555555555555555555 555555555555555555555555555555 555555555555555555555555555555 3 ( ) ( ) 5 x f x - (2)f ( 3)f 555555555555555555555555555555 555555555555555555555555555555 555555555555555555555555555555 555555555555555555555555555555 ( ) 1.44x f x - 1 ( ) 2 f ( 1.5)f 555555555555555555555555555555 555555555555555555555555555555 555555555555555555555555555555 555555555555555555555555555555 + 5x y 2 2x 555555555555555555555555555555 555555555555555555555555555555 10x y 1 1x 555555555555555555555555555555 555555555555555555555555555555 1 ( ) 3 x y 3 3x 555555555555555555555555555555 555555555555555555555555555555 0.4x y 2.5 2.5x 555555555555555555555555555555 555555555555555555555555555555 + 1 ( ) 2 x y 1y x 2 2x 555555555555555555555555555555 555555555555555555555555555555
  6. 6. Page 6 Logarithm Function f 6{( , ) / , 0 1}x x y y a a and a f 1 f 1 f 6{( , ) / , 0 1}y x y x a a and a / 7 - logay x y x a " 7 - y x a 0a 1a ( y x a % logay x - ! 7 3 8 2 23 log 8 31 5 125 5 1 3 log 125 4 1 1 81 3 1 3 1 4 log 81 6 1 log 36 y 1 6 36 y 2 6 6y 8 2y - % " ! 3log 9 10log 0.001 7 1 log 7 & 5log 125 0 0.1log 10 1 7log 1 7 3log 9y 3 9y 2 3 3y 8 2y Laws of Logarithm logay x 9 y x a n , , , 1 , 1a b x and y are positive a b 1. log 1 0a 2. log 1a a 3. log log loga a axy x y : 7
  7. 7. Page 7 4. log log loga a a x x y y ; 7 5. log n logn a ax x : 7 log 6. log log b a b x x a < 7 1 7. log logn aa x x n 1 1 8. log - log loga a a x x x 1 9. log log a x x a 11. log log n n a a x x log 10. a xa x 12. log logm n aa n x x m The prove properties : 1. log 1 0a 0 1a log 1 0a : 2. log 1a a 1 a a log 1a a : 3. log log loga a axy x y 7 loga x m loga y n m x a n y a m n xy a loga xy m n ( log log loga a axy x y : 4. log log loga a a x x y y 7 loga x m loga y n m x a n y a m nx a y loga x m n y ( log log loga a a x x y y
  8. 8. Page 8 : 5. log n logn a ax x 7 loga x m m x a ( )n m n nm x a a log nmn a x ( log n logn a ax x : log 6. log log b a b x x a 7 loga x m m x a log log m b bx a log mlogb bx a log log b b x m a ( log log log b a b x x a : 1 7. log logn aa x x n Since log log log b n na b x x a log log log b na b x x n a log1 log log b na b x x n a ( 1 log logn aa x x n =(% % Example ! 5 5 3 7 ( ) log log 7 3 a 10 10 1 ( ) log 500 log 25 2 b 6 6( ) log 9 log 4c 7 7 1 ( ) log 8 - log 14 3 d
  9. 9. Page 9 Example ! 9 9 log 512 ( ) 1 log 32 a 2 5 7( ) log 125 log 49 log 16b Solution 9 9 9 5 9 9 log 512 log 2 ( ) 1 log 2log 32 a 9 9 9log 2 5log 2 : 7 9 5 < 9log 2 2 5 7( ) log 125 log 49 log 16b > ! 9 9 log 36 ( ) 1 log 216 a 8 49 3( ) log 27 log 16 log 343b Example 23 10 10 103log ( ) 2 log logx y x y , where x and y are positive , express y in term of x 4 100 x 10 10 104log ( ) log y 1+2logx y x , where x and y are positive , express y in term of x Logarithmic Functions and Their Graphs - a 0a 1a x 9 loge x ( 0a 1a ( ) logaf x x a 0x
  10. 10. Page 10 ( 10( ) logf x x $ < 7 ( 10log x % % log x $ ( ( ) logeg x x e ? 7 ( loge x % % ln x ! ,@ =( % " + " 10 2 1.3( ) log , ( ) log ( ) logf x x g x x and h x x < 10 2 1.3( ) log , ( ) log ( ) logf x x g x x and h x x 3 A " 0 1x " 0.1 0.5 3 4 ( ) log , ( ) log ( ) logF x x G x x and H x x & < 0.1 0.5 3 4 ( ) log , ( ) log ( ) logF x x G x x and H x x 3 A " 0 1x - ( ) logaf x x 0a 1a ( ) logaf x x % 0x ( . (1,0)
  11. 11. Page 11 ( % . % % . & / 1a ( ) logaf x x - 0 / 0 1a ( ) logaf x x " - 1 , R R Example " ( ) x f x e ( ) lng x x " ( ( ) lny g x x ( ) x y f x e y x Exercise B 7 2 128 2 1 3 9 0 5 1 1 33 10 10 0.4771 10 3 3 10 0.001 B 2log 8 3 6 1 log 2 36 13log 13 1 10 1 log 10 2
  12. 12. Page 12 2 1 log 8 256 10log 300 24771 ! 3log 81 7 1 log 49 5 10log 10 loge e & ! 4 4 1 log 9 log 9 3 3log 108 log 4 log 25 1 log 125 7 ln ln x x 7log 19 7 2ln6 e 2 3log 27 log 16 4 8 log 49 1 log 343 0) 5logt x t 5 2 1 log x 5log 125x 1) 10log 2p 10log 3q p q 10log 6 10log 54 10 15 log 4 4 10log 120 C 3 27log , log y qx p 3ry x r p q ' + 0 5x D! 15 3 log 20 log log 2 2 8 8 8 12 15 log log log 0.16 5 4 1 3log5 log64 2 ln108 2ln 0.5 1 ln 45 ln125 3 3 4 5 6 7 8log 4 log 5 log 6 log 7 log 8 log 9
  13. 13. Page 13 $ ! 0x 0y 2 log3 log 4 log x x y y 2 5 101 3ln 2ln ln 5 y y x y x ) 2 4 4 4 3 2log log log , 2 x x y x y y 0x 0y y x + 2logy x 1 2 logy x 0 4x < + 104logy x 2 3 2 0x y 0 10x 9 10 2 4log 1 3 x x Example - 2 3 10 10(log 8)(log 81) 4log 400 log 256 2 3 10 10(log 8)(log 81) 4log 400 log 256 3 4 8 2 3 10 10(log 2 )(log 3 ) 4log (4 100) log 2 10 10 10(3)(4) 4(log 4 log 100) 8log 2 10 10 1012 4(2log 2 2log 10) 8log 2 10 1012 8log 2 8 8log 2 20 Example - 10 10 10log 28 log 325 log 91 10 10 10log 28 log 325 log 91 10 28 325 log 91 10 28 325 log 91 10log 100 10log 10 1 Example - 2 5 27 2 27 8 1 1 (log 16) log (log 9) log log 3 log 4 25 8 2 5 27 2 27 8 1 1 (log 16) log (log 9) log log 3 log 4 25 8
  14. 14. Page 14 3 3 3 4 2 2 3 2 2 5 23 3 2 (log 2 ) log 5 (log 3 ) log 2 log 3 log 2 2 1 2 (4) 2 3 3 3 3 8 2 1 5 Example - 10 1 2 log 16 2 10 10 1 2 log 16 2 10 6 10log 42 10 10 6 2 10 4 6 20 Exercise - 3 2log 9 log 64 - 2 2 1 log (5 log ) 2 ) 10 10 10log 28 , log 25 , log 21a b c - 10log 21 & - 6 6 6log 10 log 18 log 5 0- 5 1 3 5log 3 9 1- 2 2 2 2 5 25 125 log 30 2log 3log log 16 32 96 C- 4 3 2 2log {2log [1 log (1 log 8)]}

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