-
1
an a n
a0 = 1 1n na a =
1n na a=
1. m n m na a a += 2. ( ) m n mna a=
3. ( )m
m mnn na a= = a , a > 0, n
4. (ab)n = anbn , n n
n
a ab b
=
5. n n nab a b= , n
nn
a ab b=
6. aM = aN M = N 7. aM < aN 0 < a < 1 M > N a > 1
M < N
1 66
x +=
33
66
y =+
33
x2 4xy + y2
) -2 ) -4 ) -6 ) 30 ) 34
-
2
2 x 15 22 2 105x = 3 ) 1 2x x+ = ) 12 2x x+ + = ) 4 8x x + = 2 )
3 3x x =
-
3
4 2 55 ( 2) (2 3) 2 1x x+ + +x = 3 ) [-10, 300] ) [400, 600] )
[-64, -32] ) [250,350]
5 1 21 6
x xx x
+ =
1
3x-1 x 5x-1 x2 6 25 = 75 6 x x
-
4
7 2 2 8 121 1
2 4
x x x+ + + ) 1 1 04x
+ < ) 4 04
xx
>+
) x2 + 3x -4 < 0
8 22(( 3) 32 8
)xx x < ) (1,) ) (-2,100) ) (-10,10) ) (-, 2) 9 A = {x : 2+
2x 2x+1 23 > 0}, B = {x : 2 2 2x x 1} 1. A B 2. B A 3. AB = 4.
AB = +
-
5
10 12x -2(3x) -9(4x) + 18 = 0
x = ay ay = log x e ln x = logex 10 log x = log10x 1. lo , g 1
0a =
lo g 1a a = 2. lo , g logca ab c= b
1log logc aa b bc=
3. lo g ( ) log loga abc b c= + a log log loga a
b b cc
=
a
4. log 1loglog log
ca
c b
abb a
= =
5. loga ba b= 6. log loga b a c= b = c 7. b < c a > 1 log
loga b < a c
n
b > c 0 < a < 1 8. N = a 10n log log( 10 ) lognN a a= =
+ n log N log a log log N = M antilog M = N
-
6
11 log 3 = c 31 39
log 9 log 3 log 0.81+
) 4 43
c ) 4 43
c+ ) 8 43
c ) 8 43
c+
12 10 1 1
10 100
log 28 log 325 log 91 +
13 log 1.15 = 0.0607 log 1.16 = 0.0645 log 1183 1. 3.0607 2.
3.0618 3. 3.0625 4. 3.0638 5. 3.0645
-
7
14 1 1 12 2log (4 2 6) 2 log (2 1)x x x + + = + + 15 25 5(log
)(log )(log ) log 125y zx y z = x 16 (a,b) log(3 4) log( 1) 1x x+
> + a + b
-
8
17 A log16x + log4x + log2x < 7 B 34x-3-26(32x-3) 1 A B 18
232log 2log 9 3 0xx + = 1. 1 2. 2 3. 3 4. 4
-
9
1. 22
x +=
33
22
y =+
33
x2 4xy + y2
2. 14
8 1627 81
x =
y = 3x y
3. a n . ( ) | |nn a a= . ( ) | |nn a a=
4. 12
2 8 2 2( 2)32
+ +
1. -1 2. 1 3. 3 4. 5
-
10
5. a y = a(2x) (3,16) 1. 2 2. 3 3. 4 4. 5
6. 14
8 15125 625
x =
x
7. 43( 18 2 125 3 4)+ 1. -1000 2. 1000 3. 2 5 5 2 4. 5 2 2 5
8. 1 21 2
x 1 +
-
11
9. 1. (24)30 < 220330440
2. (24)30 < 230320440
3. 220330430 < (24)30
4. 230340420 < (24)30
10. 1 1 2 22 2
1. 32 2
2 2. 2 32 2
3. 5 3 22 2 4. 3 2 5
2 2
11. 2 13 2
4
8 (18)144 6
1. 23
2. 32
3. 2 4. 3
-
12
12. ( ) ( ) ( ) ( )2 2 31 2 2 8 1 2 2 8 + + 3 1. -32 2. -24 3.
32 16 2 4. 24 16 2
13. x 5 1. x2 25 2. |x| 5 3. x|x| 25 4. (x-|x|)2 25
14. 33 13
8
x + =
681
x
15. 8x 8x+1 + 8x+2 = 228 x
-
13
16.
1. 0.9 10 0.9 10+ < + 2. 4( 0.9)( 0.9) 0.9< 3. 3 3( 0.9)(
1.1) ( 1.1)( 0.9)< 4. 300 200125 100<
17. a x 1. a < 0 ax < 0 2. a < 0 a-x < a 3. a > 0
a-x < 0 4. a > 0 ax < a
18. 2(2 4 5) 1432
x x
19. x 2 (4 )( )
4
224
xx=
-
14
20.
1. 21000 < 3600 < 10300
2. 3600 < 21000 < 10300
3. 3600 < 10300 < 21000
4. 10300 < 21000 < 3600
21. A 1
323 1
3
2(log 1) log 4 0x x + + > A
1. (0,3) 2. (1,4) 3. (2,5) 4. (2,9)
22.
A = 22 3 7 2 11 1:
2 4
x x x
x+ + +
-
15
23. 23.
A = { }2 1 2: 3 34(15 ) 5 0x x xx + =
B = 1
5 51: log (5 125) log 6 12
xxx
+ = + +
AB
24. log2(1+tan1) + log2(1+tan2) + . . . + log2(1+tan44) 25. x y
y 1 logy2x = a 2y = b x
1. 21 (log )2
ab 2. 3. 22(log )ab 2(log )2a b 4. 22 (log )a b
-
16
z = a + bi a = Re(z) b = Im(z) z 1 1z = a + b i1 2 22 2z = a + b
i 1z z=
1a a= 2
1 2
)i
)i
)
1 2b b= 1 1z = a + b i 2 2z = a + b i 1. 1 2 1 2 1 2( ) (z z a a
b b+ = + + + 2. 1 2 1 2 1 2( ) (z z a a b b = + 3. 1 2 1 2 2 2 1 2
1 2 1 2 2 1( )( ) ( ) (z z a b i a b i a a b b a b a b i= + + = +
+
4. 1 1 1 1 2 1 2 1 2 1 22 2 2 22 2 2 2 2 2 2
( )z a b i a a b b b a a b iz a b i a b a b
+ + = = +
+ + +
1 10( 1)( 2)( 3)( 4)i i i i i+ + + +
a + bi
2 2
2
(3 2 ) (1 3 ) 1(3 ) (1 2 ) 1
i ii i
+ +
+ + ii
+
-
17
3 x + y 5 2 17 110 10
i ix yi +
= ++
4 3 24 9 10x x x 0 + = (conjugate) z a bi= + z a bi= 1. z z= 2.
1 2 1 2z z z z+ = +
3. 1 2 1 2z z z z = 4. 1 2 1 2z z z z =
5. 1 12 2
z zz z
=
z a bi= + 1 2 2
a biza b
=+
z a b= + i 2| |z a b2= + 1. | | | |z z=
-
18
2. | | | |z z= 3. 1 2 1 2| | | | |z z z z = |
4. 1 12 2
| || |
z zz z
=
5. 1 2 1 2| | | | |z z z z+ + | 6. 1 2 1 2| | | | |z z z z | 7.
1 1| | | |z z =
1 2| |zzz
= 2| |z z z=
4 3
3 2
(4 3 ) (3 4 )(3 4 ) (3 4 )
i ii i
+ +
1 1(3 2 )z
z i+
=+
z z = 29 z
| |a bi+ 2(3 4 )( )( 12 5 ) (2 4 )i a bi i i+ = =
-
19
1. 1z = + i0000
3
1. 4 22 4z z z + =2. 4 22 4z z z =3. 4 22 4z z z+ =4. 4 22 4z z
z+ + =
2. 1 1 2| | | |z z z= + = 1 2| | 3z z = 3
1 2
1 2 1 2
|11 | | 5 || |
z zz z z z
+
3. 1 2 3,, , ...z z z 1 0z =
21n nz z+ = + i
111| |z
-
20
4. 1 25 2z z+ = 5 2 1 2z i= + 11| 5 |z
5. b, k |k| 3(1 ) 107bi ki+ = +
6. 113 45 5
z i = 1 25 2z z 5+ = 2z
