Phương trình mũ và logarit

Tài liệu này là tuyển chon hơn 1500 bài tập về phương trình logarit bao gồm nhiều dạng khác nhau giúp bạn đọc có thể tự rèn luyện khả năng cũng như kĩ năng nhận dạng các loại bài tập logarit cũng như phương trìnhmũ và logarit.

2x 12x 1

2 log x3

3

xx

103 log (x2 2  x 1) log (x2 2  x 1) log (x2 4x2 1) log (x2 4 x21)

104 log (x2 23x 2) log (x2 27x 12)  3 log 32

123 log (2x ).log 22 2 2x 1 124 log5  5 x2 log2x 5  1

172 log2 x 4 log22 x4 173 log x 3.log x22  2  2 log x2 22

174 log x.log x2 3 x.log x 33  log x 3log x2  3 x

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177 3.log x3 22.log2x 1  178 xlog 43 x 22 log x3 7.xlog 23

1 3

181 log x3   2 4 log x3 182 log x.log x 32 3  3.log x3 log x2

x x 1 2

267 2.log x29 log x.log3 3 2x 1 1  

2x 12x 1

log x3

3

xx

344 log2log x3 log3log x2  346 32 l o gx  1 l o gx 1

351 Cho x0,y0và x+y = 1.Tìm giá trị nhỏ nhất của: x y

352 log (x2 2  x 1) log (x2 2  x 1) log (x2 4x2 1) log (x2 4 x21)

353 log (x2 23x 2) log (x2 27x 12)  3 log 32

354 2(log x)9 2 log x.log ( 2x 1 1)3 3  

373 log5  5 x2 log2x 5  1 374 logx 5x  log 5x

401.log2 x2  x  1   log2 x2  x  1   log2 x4  x2  1   log2 x4  x2  1 

402.2  log9 x 2  log3 x log3 2 x  1  1 

403 log2 x2  3 x  2   log2 x2  7 x  12   3  log23

404 log2xlog3xlog4xlog10 x 405 logx x  6   3

3 2

8 2

417 log x22  log x 12  1 418 2log64 x8 xlog4 x

419 log7 xlog3 x2 420 x2  3log 2x  xlog 2 5

Hoàng Ngọc Phú Page 15

447 x  log x 1 log  4x

4

1 3 log

2

1

2 8 4

2   2  

x

x x

1 3

logx3   xx2  468   1log53  log53x1 3 log511 3x 9

1 2

1

2

2

1log4log232

x

x x

22log32log

log

2 ) 10 ( log 2 log

2

2log10

2log5 5







x x

500 lg2 x3  20 lg x 1  0 501 log2 4  4 log4 2  2

x x

502

1

2 log 10

1 2

505 log2(x x2  1 ) log3(x x2  1 )  log6(x x2  1 )

506 log4(x x2  1 ) log5(x x2  1 )  log20(x x2  1 )

2

1 ) 1 (

log3 x  x  3 x  x  

508

) 3 4 4 ( log

xy g

5 1 2

1 3 ( logx3   xx2 

3 3 2

2

1 3 log log

1 )

1 3

3 2

2

1log

2

1)65(

log x  x  x  x

8 2

2

1 ) 6 5 (

)2log2

(log2  4 2 2  2  4 4 x2 

x

x x

x x

2(sinlog)sin2(sin

log

3 1

12

2log

4 1

13

540 log 3xx2 ( 3 x)  1 541 loga(1 1x)loga2(3 1x)

542 log3(2x+1)+log5(4x+1)+log7(6x+1)=3x 543 log3(x2 8x14)logx24x491

Hoàng Ngọc Phú Page 18

544 lg 1x2 3lg 1x lg 1x2 2 545 ( 2 2)

4

1log

cos 2 sin

sin 2 2 sin 3

log 7 x2 7 x2

x x

x x

(log12

112

1

12

112

1

2 2

2

2 2

x

x x

x

x

x x

x

551 log23 log25

x x

553

) 5 2 (

2 5 1

) 5 3 ( 5

3

x x

4 1

)12(12

x x

27log)

27

125()

5

3

(

5 5 )

1 ( log )

1 (

2 2 ( )

2

2

9

1 1

1 1

2

3 lg

x x x

565 log5(x2)log 5(x3 2)log0,2(x2)4

566 logx3  log3 x log x3  log3 x 0 , 5

567 2log 21 log5 2log5 1 1 0

2

5x   x  x  

568.2log29 xlog3 xlog3( 2x11)

569 3 logx4  2 log4x4  3 log16x4  0 570 log5x+log3x=log53log9225

Hoàng Ngọc Phú Page 19

576 logaaxlogxax=

a a

1 log 2 với 0<a1 577 9x + 6x = 2.4x

5 ( )

3

9x x  

)3

4(2

13

4

) 5 ( ) 2 3 ( ) 2

3

2 ) 1 5 ( 7 ) 21 5 (  x   x  x

2   2  

x

x x

2

12 3 3

1 2

6

Hoàng Ngọc Phú Page 20

621

)32(10

101)

32()

01,05

4x x  x  x

2 ) 5 3 ( ) 5

182

2

22

8

1 1

2 x  x

656 1 3x2 2x 3x2 2x

2 9 2

4    

657 10 5 1 1000 10

1 5

16

9 ) 3

4 (

) 4

3 (

Với -3<a<0

662 log (2 54) log ( 3) log3( 4)

3 1 2

3 x   x  x 663 42x + 1 54x + 3 = 5 102x

2

+ 3x - 78

- 3 = 0,01.(10x - 1)3

690 log4(x + 1)2 + 2 = log

2 4 - x + log8 (4 + x)

3

694 25x = 9x + 2.5x + 2.3x

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691 log

2 (3 - x) - log8 (x - 1)3 = 0

692 log2 (x2 + 3x + 2) - log1

4 (x2 + 7x + 12)2 = 2 + log4 3

x = 6

764 logx 5 + logx 5x - 2,25 = log2x 5 765 3logx 6 - 4log16 x = 2log2 x

766 logx 2.log2x 2 = log4x 2

2 x = 8

log x + 7

7 = 10log x + 1

816 2log5 (x + 3) = x 817 log3 (x2 - 3x - 13) = log2 x

818 log2 (1 + x) = log3 x 819 2log6 ( x + 4 x) = log4 x

820 log7 (x + 2) = log5 x 821 log3 (x2 + 2x + 1) = log2 (x2 + 2x)

822 log2 (log3 x) = log3 (log2 x) 823 3log3 (x + 2) = 2log2 (x + 1)

824 log3 (76 + 4 x) = log5 x 825 log2 (1 + 3 x) = log7 x

826 log3 (x + 1) + log5 (2x + 1) = 2 827 2x

2 - 2x 3x = 1,5

828 log4 [2log3 (1 + 3log2 x)] = 1

863 log(x - 2) = - x2 + 2x + 3 864 x + log(x2 - x - 6) = 4 + log(x + 2)

865 log(x2 - 6x + 5) = log(x - 1) + 6 - x 866 xlog2 9 = x2 3log2 x - xlog2 3

867 (1 + x)(2 + 4x) = 3.4x 868 log2 (1 + cosx) = 2cosx

869 5x + 2x = 3x + 4x 870 xlog7 11 + 3log7 x = 2x 871 log2 2

882 (2 + 2)log2 x + x(2 - 2)log2 x = 1 + x2 883 5logx - 3logx - 1 = 3logx + 1 - 5logx - 1

884 log4 (log2 x) + log2 (log4 x) = 2 885 log2 x + log3 x + log4 x = log20 x

886 log2 (x - x2 - 1).log3 (x + x2 + 1) = log6 (x - x2 - 1)

887 3x2 + 6x + 7 + 5x2 + 10x + 21 = 5 - 2x - x2 888 32x + 2 + 3x4 - 6x2 + 7 = 1 + 2.3x + 1

889 log22 (x - 1) + 3x4 - 54x2 + 247 = log2 (2x2 - 4x + 2)

Hoàng Ngọc Phú Page 27

890 2x - 1 - 2x

2

- x = (x - 1)2 891 log3 x

2 + x + 32x2 + 4x + 5 = x

920 log2 log2 x  log3 log3 x

921 log2 log3 log4 x  log4log3log2 x

922 log2 log3 x  log3log2 x  log3 log3 x 923 xlg2x23lgx4,5  102lgx

924 5lgx  50  xlg5 925 log5(x 3 )  x

2 926 3log2x  log 3x  162

x

2 1 )

1 (

cos 2

x

940 5 x  51 x  4  0

3 3 2

942

16

5 20 2

2 2

22x  2x  x  x 

943  5  24  x  5  24 x  10

2 5

3 16 5

sin 2 2 sin 3

7

x x

1 1

x

2 1

2 x    x 

xx

x

956 1  log2 x  1   logx14

Hoàng Ngọc Phú Page 29

8

1 log

1 4 log 4 4

log2 x1  2 x   1/ 2

2 sin log sin

2 sin

log

3 1

x

9 3

3 2

2

1 log

2

1 6

log

3 2

962 log3x7 9  12 x  4 x2  log2x3 6 x2  23 x  21   4

963 15x  1  4x 964 2  32  1

x x

2

) 1 (

1 2 log 2

x

xx

x x

x

2

2 2

2

2 1 1

2    16 2

982 2x  2x 1  2x 2  3x 3x 1  3x 2

983 2 3x x 1.5x 2  12 984 (x2  x 1)x21 1

985 ( x  x )2 x 2  1 986 (x22x2) 4 x 2 1

1000 22x 1  32x  52x 1  2x  3x 1  5x 2 1001 log x5  log x5  6   log x5  2 

xlog 2x  5x  4  2

2log xlog x.log ( 2x 1 1) 1079 log2x log3x log2x.log3x

1080 log5x log3x log 3.log 2255 9

Hoàng Ngọc Phú Page 33

log (x x  1).log (x x   1) log (x x  1)

1102 3 log2xlog (8 ) 1 02 x   1103 1 log (  2 x  1) logx14

1 6 5 log9 x2  x 2  3 x  3 x

1172 log2 x + log3 x + log4 x = log5 x 1173 log4(log2 x) + log2(log4 x) = 2

3 2 2 1 4

8log

1183 log2 x.log3 x = 2log2 x + 3log3 x –6

1184   1log53  log53x1 3 log511 3x  9

2 6

3 x  x  x  x  x  x

x

x x

4 4

log

2 10

log 2 log

2

1189 log2x42 log 22x14log23

2

2 4

2

4 x 9 3 2 log x 3 10 log x 3 log

2

1 1 x log 1 x

log3 3  3   3 

1193 log 5 x log 5 x log 5 x log  x 5 log  x 1 log 5 x

2 2

2 2

1 x 2

2

3 2 2 1 4

8log

1200 log2 x.log3 x = 2log2 x + 3log3 x –6

1201   1log53  log53x1 3 log511 3x 9

x

1202 2 2 6 3 2 3 1 2 2 6 3

2 6

x

x x

4 4

log

2 10

log 2 log 2

Hoàng Ngọc Phú Page 36

1210.log2log2x log3log3x 1211 log2log3log4x log4log3log2x

1212 log2log3x log3log2x log3log3x 1213 log2logx3  log3logx2

sin 2 2 sin 3 log 7 x2 7 x2

x x

x x

11

x

1229 log 4 4 log 2 1 3

2 1

1230 log39 1 4 3  2 3  1

x x

14log.44

log2 x1 2 x  1/ 2

2 sin log sin

2 sin

log

3 1

x

9 3

3 2

2

1 log

2

1 6 5

cos 2 sin

x

165log9 x2 x 2  3 x  3 x

2

1123

3 2

1)13(

2 log 2

2

1)2ln(

)

2

8 2

2

log x   x x

Hoàng Ngọc Phú Page 38

1276

2

1)]}

log31(log1[log2

log4 3  2  2x  1277 logcosx 4 logcos 2x 2  1

2

1)58lg(

)8lg(x3   x  x2  x 1279 (  2 )log39( 2)  9 (  2 )3

1287 log2( x  3log6x)  log6 x

1288 log2(x2x1)log2(x2x1)log2(x4x21)log2(x4x21)

1289 log2(x2 3x2)log2(x27x12)3log23

1294 log2(x x2  1 ) log3(x x2  1 )  log6(x x2  1 )

1295 log4(x x2  1 ) log5(x x2  1 )  log20(x x2  1 )

2

1)1(

1297

)344(log

xy g xy

5 1 2

2

13loglog

3

1303 log (4 4) log (2 1 3)

2 1

Hoàng Ngọc Phú Page 39

1304 log3x7( 9  12x 4x2)  log2x3( 6x2 23x 21 )  4

1305 2 log ( 1 ) 2 log 1 ) 1 3 ( log 2 3 2       x x x 1306 log log3( 9x  6 )  1 x

1307 log3( 9 1 4 3  2 )  3  1 x x x 1308 log 2 6 log 2 4 2 22 2.3 log 4 xx  x

1309 2 9 3 3 2 27 log ( 3 ) 2 1 log 2 1 ) 6 5 ( log x  x  x  x

1310 3 8 2 2 4( 1) 2 log 4 log (4 ) log x   x  x

1311 2 3 2 3 ( 1 ) log 2 log x x  x xx

1312 log2(x2+x+1)+log2(x2-x+1)=log2(x4+x2+1)+log2(x4-x2+1) 1313 (  1 ) log53  log5( 3x1 3 )  log5( 11 3x  9 ) x

1314 log 3 2 1 log 2 1 ) 6 5 ( log9 x2  x 2  3 x  3 x

1315 log 2 14log16 3 40log4 0 2    x x x x x x

1316 cos2 ) 0 2 (sin log ) sin 2 (sin log 3 1 3 x x  x x 

1317 log 3xx2 ( 3 x)  1 1318 loga(1 1x)loga2(3 1x)

1319 log3(2x+1)+log5(4x+1)+log7(6x+1)=3x

1320 log3(x2 8x14)logx24x491

1321 lg 1x2 3lg 1x lg 1x2 2 1322 ( 2 2) 4 1 log 2 1 x  x  x

1323 log ( 2 2 2) log2 3( 2 2 3) 3 2 2  x  x   x  x

1324 log 2 cos 2 sin sin 2 2 sin 3 log7 x2 7 x2 x x x x    

1325 9 11 ) 2 2 ( log 1 2 1 1 2 1 1 2 1 1 2 1 2 2 2 2 2                x x x x x x x x x x

1326 ) 5 2 ( log 2 2 5 1 ) 5 3 ( 5 3 1 x x x x      1327

) 2 7 1 ( log 2 4 1 ) 1 2 ( 1 2 1 x x x x     

Hoàng Ngọc Phú Page 40

1328 2log5x2  21log5x 2log5x1 0

1329

243log

27log)

27

125()

5

3(

5 5 )

1 ( log )

1 ( log

2 2 ( )

2 2

1 1

1 1

2

3 lg

5

2 ( log0 , 25(x25x8) 

x x

2

1

2 1

2

1338 ( 2  2  2 ) 9 2 3 2  2  2

x x

x

x x x

2 5 6

2

2323

6

1 2

1 2

x

x x

1442 x  x2  x

1 log 2

)4(log2)3(log

1

2 4 1

) 1 log(

1

2 )

1 ( log

1

) 1 log(

x

1445 logx3x(x24)log4x26(x24)

1446

4log

1log.log2log

2 2 2

3 3

x x

x

42log)21(log

2 1

) 10 log(

)

10

log(

3.26

Hoàng Ngọc Phú Page 41

2

1 log

2

1 ) 6 5 (

log9 x2  x 2  3 x  3x

1454 log2x(6x25x1)log3x(4x24x1)20

8 2

log2x 4x  1457 log22log24x3

x

2 log

1 ) 1 3

x

1459 log4(x x2  1 ) log5(x x2  1 )  log20(x x2  1 )

1 2 cos

1212

x x

x x

x x

1472 2x21.log (2 x2   1) 4x1.(log2 x   1 1)

1473 lnx1lnx3lnx7 1474 4 3

log 2 4 log logx  x  x

40log

11log

1479 6 4 13 6 6 9 0

1 1 1

8  

log 2

1 log

4

1

3 3

2

21

2

8

1 1 2

5x x  x x

1491 42x 23x1 2x3 16  0

1492 3cosx 2cosx  cosx

1493 4 log 3x 2 log 3x  2x 1494 3cos2x 3sin2x  2x 2 2 2

2 3 3

x

12

12

x x x x

2

564

483627

161232

2

1

1 2 log 2 6

1 2

2 2

2

2 2

3

32

12

x x

x x

x

x

x