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ECUATII LOGARITMICE RECAPITULARE Profesor Mihaila Adriana

Obiective :sa rezolve ecuatii logaritmicefolosind proprietatile logaritmului

- sa rezolve ecuatii logaritmice folosind ecuatiade gradul II si I

Sa se rezolve ecuaiile logaritmice (care continbaze diferite) :

1) 2log 9 log 2 10xx + = 2) 4log ( 12) log 2 1xx + =

3 7 51 log 5 log log 35 log 5x xx+ =

4) ( )2

2

1 2

2

log 4 log 8

8

xx

+ =

Exercitii:Rezolvati ecuatiile logaritmice (cecontin logaritmi in aceeasi baza):

1) ( ) ( )4 25 53

log 5 log 252

x x+ + + =

2) ( )3 3 3log 2 log log 8x x + =

3) lg( 9) 2 lg 2 1 2x x + =

4) 3 3 3log (5 2) 2 log 3 1 1 log 4x x + =

5) 2 22 3 7

log 1 log

1 3 1

x x

x x

=

6) ( ) ( )24 4log 1 log 5 0x xx x+ + =

7) log3x = log3(x+6) log3(x+2);

Rezolvati ecuatiile logaritmice cu ajutorul

formulei2log log 0x x + + =

1)2

lg 4 lg 3 0x x + = 2) 2lg lg100x x=

3)2 2lg lg 3x x+ = 4) ( ) ( )

2

2

2lg 1 1

lg 1x

x+ =

+

5) ( ) ( ) 32 2 2 2log 3 log 2 log 1 log 2 0x x x + + =

6) ( ) ( )4 22 21 13

log 4 log 4 3,6 010 10

x x + =

7) ( )5 52 log 2 log 25 2 5x x+ + =

Rezolvati ecuatiile logaritmice (cu ajutorul

formulei1

loglog

a

b

ba

=

1) 33

log log 3

2

xx =

2) 4 163 log 4 2 log 4 3log 4 0x x x+ + = 3)

( )2 3log log log 11a a ax x x+ + =

4)2

2

4 16 4log 2) log 4 log 4 log4

x x

xx

+ =

5) 2 2(log 2)(log 2)(log 4 ) 1x x x =

5)2 2 2

3 9 27

49log log log

9x x x+ + =

6) ( )4 2 2 12 2

log 2(log 2) log 2 0

log 2x

x

x

+ =

7) 16 23 log 16 4 log 2 logx x x =

8) 3 133

log log log 4x

x x x+ =

9) ( ) ( )2 22 1 5 2log 5 8 4 log 1 4 4 4x xx x x x+ + + + + =

Sa se rezolve ecuatiile logaritmice (care continexpresiide forma logaxx

1) lg 21000xx x= 2) ( )lg

1000x

x x=

3)2log4

3

x

x =

. Cea mai mica radacina.

4) 25 log log 2 8xxx = . Cea mai mare radacina.5) 22 log 256xx + = 6) 3log5 243xx =7)

22lg 310xx x= 8) 4 4log 3(log 3)2x xx +=

9)3log 3 log

9 27x x

x = 10)5log 5 log

25 125x x

x =11)

( ) ( ) 33loglog 1 21 2

xxx x x

+ + + =Sa se rezolve ecuatiile mixte (exponential-logaritmice) :

1) ( )23log 3 3 63x x x = 2)

( )2 2 17log 3 3 7 2x x x x + + =

3) ( )6 61 log 2 log 2 1xx = +

4) ( ) ( )32 2log 4 1 log 2 6x xx ++ = +

5) ( )4

log 3

2log 9 2 16xx

=6) ( )1 12 2log 9 7 2 log 3 1

x x + = + +

7) ( ) ( )13 3log 3 1 log 3 3 6x x+ =

8) ( ) ( )12 2log 4 1 log 4 4 1x x+ =

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9)

11

1 lg3 lg 2 lg(27 3 )2

x

x

+ + =

10) ( )25 53log 2 2 log 3 5x xx + =

Sa se rezolve ecuatiile logaritmice (care continradicali) ;

1) 2 2log 2log 2x x=

2) ( )2 5lg 3 4 1 lg 2x x+ =

3) 3 35 log log 9 4 0x x = Cea mai micaradacina

4) ( )2 5log 2 2 5 2 3 0,5x x x+ + =

5)5log 5 log 1x x x =

6) 2 25 5log log 5 22

xx + + =

Vreti mai mult ?

1)2log log .... log log log ....logn n

n

a xa a x xx x x a a a+ + + = + +

2)2 2log log .... log log 2 log .... logn n

n n

a xa a x xx x x a a n a+ + + = + + +

3) 2 3 12 3 1log log .... log log log .... log nn

n na a a x x xx x x a a a+

++ + + = + + +

unde a, x ( ) { }0, 1