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Bµi tËp ph−¬ng tr×nh logarit, mò.
Bµi 1 Gi¶i c¸c ph−¬ng tr×nh sau a) 22x−3 = 4x
Bµi 6 Gi¶i ph−¬ng tr×nh sau x
57 = 75
2 +3x−5
1
3
b) 9x − 2x+ 2 = 2x+ 2 − 32x−1
Bµi 7 Gi¶i c¸c ph−¬ng tr×nh sau
c) 9x − 3x − 6 = 0 d) 9x
2 −1
− 36.3x
2 −3
a) 2 −
√ x 5 3 +
b) 4x+
√
x2 −2
c)
√ x−10 3 − 84 = 0
10
− 5.2x−1+
√
x2 −2
2
x
2
+ 9cos
b) 4cos 2x + 4cos
2
√
3−
√ x √ √ x √ 2 + 3 + 2 = ( 5)x x
d) 3x − 4 = 5 2
=6
e) 125x + 50x = 23x+1
Bµi 3 Gi¶i ph−¬ng tr×nh sau a) 9sin
√ x √ x 3 + 2 + 3 = 4x
b) 3x + 4x = 5x
+3=0
Bµi 2 Gi¶i c¸c ph−¬ng tr×nh sau a)
x
Bµi 8 Gi¶i c¸c ph−¬ng tr×nh sau
x
= 10
a) 8.3x + 3.2x = 24 + 6x
x
=3
b) 4x
Bµi 4 Gi¶i c¸c ph−¬ng tr×nh sau √ tan x √ tan x + 3−2 2 =6 a) 3 + 2 2 p p √ cos x √ cos x b) 7+4 3 + 7−4 3 =4 √ x √ √ x √ c) 2+ 3 + 7+4 3 2− 3 = 4(2+ 3) Bµi 5 Gi¶i c¸c ph−¬ng tr×nh sau
2 −3x+2
+ 4x
2 +6x+5
= 42x
2 +3x+7
c) 52x = 32x + 2.5x + 2.3x Bµi 9 Gi¶i ph−¬ng tr×nh sau 2x+1 − 4x = x − 1 Bµi 10 Gi¶i ph−¬ng tr×nh sau 2
6log6 x + xlog6 x = 12
a) 18.4x − 35.6x + 12.9x = 0
Bµi 11 Gi¶i ph−¬ng tr×nh sau
b) 25x + 10x = 22x+1
5x .8
x−1 x
= 500
log(100x2 )
c) 4log(10x) − 6log x = 2.3
√ x √ x d) 5 − 21 + 7 5 + 21 = 2x+3
3
Bµi 12 Gi¶i ph−¬ng tr×nh sau x2 +1 x2 − x − 1 =1
+1
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Bµi 13 Gi¶i ph−¬ng tr×nh
Bµi 21 Gi¶i ph−¬ng tr×nh
2 x − 3 x −x = |x − 3|2
log2 (x2 + 3x + 2) + log2 (x2 + 7x + 12) = 3 + log2 3
Bµi 14 Gi¶i ph−¬ng tr×nh 25x − 2(3 − x).5x + 2x − 7 = 0
Bµi 22 Gi¶i ph−¬ng tr×nh log9 (x2 − 5x + 6)2 =
Bµi 15 Cho ph−¬ng tr×nh m9x + 3(m − 1)3x − 5m + 2 = 0 T×m m ®Ó ph−¬ng tr×nh ®· cho: a) cã nghiÖm b) Cã hai nghiÖm tr¸i dÊu c) Cã hai nghiÖm ph©n biÖt ©m. d) Cã hai nghiÖm x1 , x2 tháa ®iÒu kiÖn 0 < x1 < 1 < 2 < x2 Bµi 16 Gi¶i c¸c ph−¬ng tr×nh sau a) log4 (x + 3) − log4 (x − 1) = 2 − log4 8 2
b) log21 (4x) + log2 ( x8 = 6)
1 x−1 log√3 ( ) 2 2
+ log3 |x − 3| Bµi 23 Gi¶i ph−¬ng tr×nh log 2 + log2 4x = 3 x
Bµi 24 Gi¶i ph−¬ng tr×nh √ log4 (x+1)2 +2 = log√2 ( 4 − x)+log8 (4+x)3 Bµi 25 Gi¶i ph−¬ng tr×nh log2 x + log3 x + log5 x = 1 Bµi 26 Gi¶i ph−¬ng tr×nh
2
c) log√5 (4x − 6) − log5 (2x − 2)2 = 2 Bµi 17 Gi¶i ph−¬ng tr×nh
log2 x log3 x = 1 Bµi 27 Gi¶i ph−¬ng tr×nh
log3 (3x − 1) log3 (3x+1 − 3) = 6 Bµi 18 Gi¶i ph−¬ng tr×nh
1 log4 2 log3 (1 + log2 (1 + 3 log2 x)) = 2 Bµi 28 Gi¶i ph−¬ng tr×nh
x
x + log2 (9 − 2 ) = 3 Bµi 19 Gi¶i ph−¬ng tr×nh √ log2 (x + 1)2 + log2 ( x2 + 2x + 1) = 6 Bµi 20 Gi¶i ph−¬ng tr×nh log2 (x2 + x + 1) + log2 (x2 − x + 1) 4 x2 + 1) = log2 (x4 − x2 + 1) + log2 (x4 +
2 log 1 (4 − x) 1 4 + =1 log6 (x + 3) log2 (x + 3) Bµi 29 Gi¶i ph−¬ng tr×nh log7 x = log3 (2 +
√
x)
Bµi 30 Gi¶i ph−¬ng tr×nh x + xlog2 3 = xlog2 5
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Bµi 31 Gi¶i ph−¬ng tr×nh
Bµi 34 T×m m ®Ó ph−¬ng tr×nh log3 (x2 + 4mx) + log 1 (2x − 2m − 1) = 0
logx (x + 1) = log3 2
3
cã duy nhÊt nghiÖm.
Bµi 32 T×m m ®Ó ph−¬ng tr×nh lg(x2 + mx) = lg(x + m − 1) Bµi 33 Gi¶i ph−¬ng tr×nh √ √ (2 + 2)log2 x + x(2 − 2)log2 x = 1 + x2
Bµi 35 Gi¶i ph−¬ng tr×nh √ log2 (1 + x) = log3 2 Bµi 36 Gi¶i ph−¬ng tr×nh 25log0,2 (sin
cã duy nhÊt nghiÖm.
2
x+6 sin x cos x+5)
1 36
=
Bµi tËp bÊt ph−¬ng tr×nh logarit, mò.
Bµi 1 Gi¶i bÊt ph−¬ng tr×nh
Bµi 6 Gi¶i bÊt ph−¬ng tr×nh x
4x+1 − 16x < 2 log4 8
3x+1 − 22x+1 − 12 2 < 0
Bµi 2 Gi¶i bÊt ph−¬ng tr×nh √ √ x−1 ( 5 + 2)x−1 ≥ ( 5 − 2) x+1
Bµi 7 Gi¶i bÊt ph−¬ng tr×nh
Bµi 3 Gi¶i bÊt ph−¬ng tr×nh
Bµi 8 Gi¶i ph−¬ng tr×nh
32x √ 9 x−1
− 8.3x−
√
x−1
−9>0
Bµi 4 Gi¶i bÊt ph−¬ng tr×nh 2 +1
4x2 + x.2x
2
2
+ 3.2x > x2 .2x + 8x + 12
Bµi 5 Gi¶i bÊt ph−¬ng tr×nh 2x−x2 +1
25
2x−x2 +1
+9
√
8.3
√ x+ 4 x
+9
√ 4
x+1
√
≥9
x
log3 (x2 + x + 1) − log3 x = 2x − x2 Bµi 9 Gi¶i bÊt ph−¬ng tr×nh √ √ log2 (x − x2 − 1) log3 (x + x2 − 1) √ = log6 (x − x2 − 1) Bµi 10 Gi¶i ph−¬ng tr×nh
2x−x2
≥ 34.15
logx (cos x−sin x)+log 1 (cos x+cos 2x) = 0 x
5
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Bµi 11 Gi¶i bÊt ph−¬ng tr×nh log2
Bµi 21 Gi¶i bÊt ph−¬ng tr×nh √ log5 ( 3x + 3) logx 5 > 1
x2 + 8x − 1 ≤2 x+1
Bµi 22 Gi¶i bÊt ph−¬ng tr×nh Bµi 12 Gi¶i bÊt ph−¬ng tr×nh log2
log2 x + log3 x < 1 + log2 x log3 x
15 log 1 (2 − ) ≤ 2 2 16 x
Bµi 23 Gi¶i bÊt ph−¬ng tr×nh logcos x logsin x (sin x + cos 2x) > 0
Bµi 13 Gi¶i ph−¬ng tr×nh log2 (2x + 1) + log3 (4x + 2) = 2
Bµi 24 Gi¶i vµ biÖn luËn bÊt ph−¬ng tr×nh
Bµi 14 Gi¶i c¸c bÊt ph−¬ng tr×nh sau
xloga x+1 > a2 x
a) log2 (2x + 1) + log3 (4x + 2) ≤ 2 b) log2 (2x +1)+log3 (4x +2)+log5 (3x +4) ≤ 3 Bµi 15 Gi¶i bÊt ph−¬ng tr×nh (4x − 12.2x + 32) log2 (2x − 1) ≤ 0 Bµi 16 Gi¶i bÊt ph−¬ng tr×nh √ √ log3 x2 − x − 6+log 1 x − 3 > log 1 (x+2) 3
3
Bµi 17 Gi¶i bÊt ph−¬ng tr×nh p log9 (3x2 + 4x + 2)+1 > log3 (3x2 +4x+2) Bµi 18 Gi¶i bÊt ph−¬ng tr×nh
Bµi 25 Cho a ≥ 1, b ≥ 1, chøng minh r p p a+b ) log2 a + log2 b ≤ 2 log2 ( 2 Bµi 26 Cho a > 0, b > 0, chøng minh log 1 a + log 1 b ≥ 2 log 1 ( 2
2
2
a+b ) 2
Bµi 27 Gi¶i ph−¬ng tr×nh 2sin x + 2cos x = 21−
√ 2 2
Bµi 28 Gi¶i ph−¬ng tr×nh 2 log3 cot x = log2 cos x
1 log√2 (x−2)+log2 |x−4|−1 Bµi 29 T×m m ®Ó mäi nghiÖm cña bÊt ph−¬ng 2 tr×nh 1 1 Bµi 19 Gi¶i bÊt ph−¬ng tr×nh ( )x > 3 3 4x − 2 1 ®Òu tháa m·n bÊt ph−¬ng tr×nh logx2 ≥ |x − 2| 2 x2 + (1 − 2m)x + m − 5 < 0 Bµi 20 Gi¶i bÊt ph−¬ng tr×nh Bµi 30 T×m m ®Ó bÊt ph−¬ng tr×nh |x − 5| 1 4x + 2m.2x − 3m + 4 > 0 tháa logx3 ≥ 6 6x 3 a) ∀x ∈ R b) ∀x > 0 c) ∀x ∈ [0, 1] log4 (x2 −7x+12)2 <
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Bµi tËp tù rÌn luyÖn.
Bµi 1 Gi¶i c¸c ph−¬ng tr×nh sau
Bµi 4 Cho ph−¬ng tr×nh logarit sau q 2 log3 x + log23 x + 1 − 2m − 1 = 0
a) log4√(x + 1)2 + 2 = log√2 4 − x + log8 (4 + x)3
a) Gi¶i ph−¬ng tr×nh khi m = 2.
b) lg4 (x − 1)2 + lg2 (x − 1)3 = 25 c) log2
√ 3
p x + 3 log2 x =
b) T×m m ®Ó ph−¬ng tr×nh √ cã Ýt nhÊt mét nghiÖm thuéc ®o¹n [1, 3 3 ]
4 3
d) logx2 (2 + x) + log√2+x x = 2 Bµi 2 Gi¶i c¸c ph−¬ng tr×nh sau a) log2 (4x + 4) = x − log 1 (2x+1 − 3) 2
b) log3 (9x+1 + 4.3x − 2) = 3x + 1 c) log3
x2 + x + 3 = x2 + 3x + 2 2x2 + 4x + 5
d) | ln(2x − 3) + ln(4 − x2 )| = | ln(2x − 3)| + | ln(4 − x2 )| Bµi 3 Gi¶i c¸c ph−¬ng tr×nh sau a) log1995 (tan x) = cos 2x b) logx
3x + 2 x+2
>1
c) |1 + logx 2009| < 2 d) logx (log3 (9x − 72)) ≤ 1
7