Exponential and logarithm function

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

! ( ) 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

" 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

( 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

( ) 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

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

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

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

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( ) 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)

( % . % % .
& / 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

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

$ ! 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

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)]}

Exponential and logarithm function

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Exponential and logarithm function