Chuaồn bũ cho kyứ thi ẹaùi hoùc

Vi bi toỏn v phng trỡnh
logarit khỏc c s
Hunh c Khỏnh 0975.120.189
Gii tớch H Quy Nhn

Descartes

Phng trỡnh logarit vi c s khỏc nhau luụn l vn gõy khú d cho hc sinh khi gp phi
trong cỏc thi. Hc sinh thng lỳng tỳng khi bin i, gp khú khn a v cựng c s hoc a v
cỏc phng trỡnh c bn. Tụi vit bi xin úng gúp vi bi mu v vn ny, nú c dựng cỏc phng
phỏp:
i c s, t n ph a v phng trỡnh m, bin i tng ng, ỏnh giỏ hai v.

Vớ d 1. Gii phng trỡnh:
log2 x + log 3 x + log 4 x = log20 x .

iu kin: x > 0 .
Vi iu kin trờn phng trỡnh tng ng
log2 x + log 3 2. log2 x + log 4 2. log2 x = log20 2.log2 x

log2 x ( 1 + log3 2 + log4 2 log20 2 ) = 0
log2 x = 0

(do 1 + log 3 2 + log 4 2 log20 2 0 )

x = 1 (tha món).
Vy phng trỡnh cú nghim x = 1 .

Vớ d 2. Gii phng trỡnh:

t

(*)

t

3
1
2
Hm s y = + 13 + 3 l tng ca cỏc hm nghch bin nờn y nghch bin,
4
4
4
hm y = 1 l hm hng. Do ú phng trỡnh (*) cú nghim duy nht.
3

3

3

3
1
2
Ta cú: 1 = + 13 + 3 . Suy ra phng trỡnh (*) cú nghim t = 3 .
4
4
4
Vi t = 3 x = 23 = 8 (tha món).
Vy phng trỡnh cú nghim duy nht x = 8 .
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= 1 .
+
2
2

(*)

t

t
3
1

l tng ca cỏc hm nghch bin nờn y nghch bin, hm
Hm s y = +
2
2
y = 1 l hm hng. Do ú phng trỡnh (*) cú nghim duy nht.
2

2
3
1

= 1 . Suy ra phng trỡnh (*) cú nghim t = 2 .
Ta cú: +
2
2

Vi t = 2 x = 32 = 9 (tha món).

t

(3)

2
1
Hm s y = + l tng ca cỏc hm nghch bin nờn y nghch bin, hm y = 1
3
3
l hm hng. Do ú phng trỡnh (3) cú nghim duy nht.
1

1

2
1
Ta cú: + = 1 . Suy ra phng trỡnh (3) cú nghim t = 1 .
3
3
x = 1 3
1
2
(tha món).
Vi t = 1 u = 2 = 2 x + 2x = 2
x = 1 + 3
Vy phng trỡnh cú nghim x = 1 3; x = 1 + 3 .
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Chuaồn bũ cho kyứ thi ẹaùi hoùc


t

1
1
Hm s y = 625 + 2 l tng ca cỏc hm nghch bin nờn y nghch bin, hm
15
3
y = 3 l hm hng. Do ú phng trỡnh cú nghim duy nht.
2

2

1
1
Ta cú: 3 = 625 + 2 . Suy ra phng trỡnh cú nghim t = 2 .
3
15
Vi t = 2 x + 1 = 32 x = 8 (tha món).
Vy phng trỡnh cú nghim duy nht x = 8 .

Cỏch khỏc: Kim tra x = 8 l nghim ca phng trỡnh.
Nu x > 8 thỡ

log 3 ( x + 1 ) > log3 ( 8 + 1 ) = 2
log5 ( 3x + 1 ) > log5 ( 3.8 + 1 ) =


log ( x + 1 ) + log ( 3x + 1 ) > 4 .
3

Vi iu kin trờn phng trỡnh tng ng
log2 ( x 1 )( x 4 ) + log5 ( x 4 ) = 1 + log2 5 ( x 1 )
2

log2 ( x 1 ) + log2 ( x 4 ) + log5 ( x 4 ) = 1 + log2 5 + log2 ( x 1 )
log2 ( x 4 ) + log5 2. log2 ( x 4 ) = 1 + log2 5
( 1 + log5 2 ) log2 ( x 4 ) = 1 + log2 5
log2 ( x 4 ) =

1 + log2 5
1 + log5 2

log2 ( x 4 ) = log2 5
x4 = 5
x = 9 (tha món).
Vy phng trỡnh cú nghim x = 9 .

Vớ d 7. Gii phng trỡnh:
log 3x + 7 ( 4x2 + 12x + 9 ) + log2x + 3 ( 6x 2 + 23x + 21 ) = 4 . (1)
3
< x 1 .
2
Vi iu kin trờn phng trỡnh tng ng

iu kin:

2

log 3x + 7 ( 2x + 3 ) + log2x + 3 ( 3x + 7 )( 2x + 3 ) = 4


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x = 2 ( loai )

.
3x + 7
x = 1

4


Chuaồn bũ cho kyứ thi ẹaùi hoùc
Vớ d 8. Gii phng trỡnh:
log x ( x + 1 ) = lg 2 .

iu kin: 0 < x 1 .
Nu 0 < x < 1 thỡ x + 1 > 1 , ta cú
log x ( x + 1 ) < log x 1 = 0 = lg1 < lg 2 .
Nu x > 1 thỡ x + 1 > x , ta cú
log x ( x + 1 ) > log x x = 1 = lg10 > lg 2 .
Vy phng trỡnh vụ nghim.
Vớ d 9. Gii phng trỡnh:
log2 x + log 3 ( x + 1 ) = log 4 ( x + 2 ) + log5 ( x + 3 ) .
iu kin: x > 0 .
Kim tra x = 2 l mt nghim ca phng trỡnh.
Nu 0 < x < 2 thỡ
x
x+2
x +1 x + 3
>

Tng t cho trng hp x > 2 , ta c
log2 x + log 3 ( x + 1 ) < log 4 ( x + 2 ) + log5 ( x + 3 ) .
Vy phng trỡnh cú nghim duy nht x = 2 .
Vớ d 10. Gii phng trỡnh:
log2 ( log 3 x ) = log 3 ( log2 x ) .

iu kin: x > 1 .
log 3 x = 2t (1)
log2 ( log3 x ) = t
.
t: log2 ( log 3 x ) = log 3 ( log2 x ) = t . Khi ú

log 3 ( log2 x ) = t
log2 x = 3t (2)


t
t
t
2
log 3 x
log x 2 2
2
Suy ra:
= t
= log 3 2 = t = log 2 ( log 3 2 ) .
3
log x
log 3 3
3


3. log 3 ( x 2 − 1 ) = log2 x .

4. log2 ( x + 1 ) − log3 ( x + 1 ) = 0 .

x+

x = log4 x .

2

3

5. log6 ( x 2 − 2x − 2 ) = log5 ( x2 − 2x − 3 ) . 6. log2 x = log 3 x .
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