DOÃN XUÂN HUY-THPT Ân Thi-Hưng Yên PHƯƠNG TRÌNH, HỆ PHƯƠNG TRÌNH, BẤT PHƯƠNG TRÌNH MŨ VÀ LÔGARÍT I.Phương trình, bất phương trình mũ : 1/ Đưa về cùng một cơ số hoặc hai cơ số: 1 2 8 1 3 1 2 3 4 1/ 3 0,2 1/ 2 4 ;2/3 3 3 3 750;3/5 .8 500 (5.2 ) 1 3; 2 x x x x x x x x x x x x x log − − + − + − − − − = + − + = = ⇔ = ⇒ = 1 1 1 1 4/( 5 2) ( 5 2) 1 ( 2; 1) (1; ) 1 x x x x x x x − − + − + ≥ − ⇔ − ≥ − ⇒ ∈ − − ∪ +∞ + ( ) 1 2 1 2 9/ 4 5/9 9 9 4 4 4 9 .91 4 .21 9/ 4 21/91 log (21/91) x x x x x x x x x x + + + + + + < + + ⇔ < ⇔ < ⇔ < 2 2 3 1 2 6/ 2 .4 256;7/ 2 .5 0,01;8/ 2 . 3 216;9/(3 3 3 ) (1/81) ;10/ 2 .3 .5 12 x x x x x x x x x x x+ − − = = = = = 2 2 3 1 1/ 2 1 4 2 2 1/(3 1) 2 1 3 11/ 2 5 ;12/8 36.3 ;13/1 5 25;14/ 2 2 ;15/( 10 3) ( 10 3) x x x x x x x x x x x x x − + − − − − − + + − + = = < < ≥ + < − 2/ Đặt ẩn phụ: 2 2 2 2 2 2 1 2 1/(7 4 3) 3(2 3) 2 0( 3/ 2 0);2/(3 5) (3 5) 2 0 x x x x x x x x t t − − + − + − − + = − + = + + − − ≤ 3 3( 1) 1 2 4 4 ( 1/ 2 0);3/ 2 6.2 1/ 2 12/ 2 1( 2 2 );4/3 8.3 9.9 0 x x x x x x x x x x t t t − − + + + + − ≤ − − + = = − − − > ( chia 2 vế cho 2 3 x ); 2 2 2 1 2 3 2 1 5/ 4 5.2 6 0;6/ 4 7.4 2 0 x x x x cosx cosx+ − − + − + + − − = − − = ; 2 2 2 2 1 4 1 7/27 6.64 6.36 11.48 ;8/ 2 2 2 ;9/( 5 2 6 ) ( 5 2 6 ) 10 x x x x x x x x x x+ − − − = − + = + + − = 2 1 2 2 7 2.3 2 2 4 1 1 10/ 6.(0,7) 7;11/ 1 1 ;12/ 3. 12 100 3 2 1 3 3 x x x x x x x x x x t t + + − −       = + ≤ ≤ + =  ÷  ÷  ÷ − −       2 2 2 2 2 1 2 2 2 3 6 3 5 2 13/9 9 10;14/ 2 9.2 2 0;15/ 2 15.2 2 ;16/ 9 3 3 9 sin x cos x x x x x x x x x x x x+ + + + − − + − + + = − + = + < − > − 2 1 / 2 3 1 17/ 25 10 2 ;18/ 4 2.6 3.9 ;19/ 4.3 9.2 5.6 ;20/125 50 2 x x x x x x x x x x x x+ + + = − = − = + = . 3/ Sử dụng tính đơn điệu của hàm số: / 2 1 1 1/ 4 2 4 1/ 2 1 3 ;2/ 2 3 6 1;3/(2,5) (0,4) 2,9;4/3 2 12;5/ 2 6 x x x x x x x x x x x + + + + = + + > − + = + > = − 1 DOÃN XUÂN HUY-THPT Ân Thi-Hưng Yên 2 2 1 2 3 2 6 10 2 6/ 2 2 ( 1) ;7/ 2 8 14;8/3 6 6;9/3 5 6 2 x x x x x x x x x x x x x x − − − − + − = − = − + − = − + − + = + 2 3 2 2 2 2 1 10/3 (3 10).3 3 0;11/3.25 (3 7).5 2 0;12/ (3 2 ) 2 2 0 x x x x x x x x x x x x − − − − + + − + − = + − + − = − − + − = 2 2 1 2 2 1 1 2 3 3 2 13/3 3 2 2 6 2 6;14/ 2 3 5 2 3 5 ;15/ 0 4 2 x x x x x x x x x x x x x x x − − − − − + + + + − − + − − = − + + + = + + ≥ − . 2 2 ( ) ( ) 2 1 16/ 4 4 8 12 1/ 2( 3/ 4);17/ 4 2 ( ) 2 0( ;0) x sin x cos x sinx sinx x x cos xy k π π π + + = − + − ± − + = 18/ ( ) 2 2 2 2 2 2 1 2 2 1 .2 0,5. 2 2 2 2 0 0;0,5 cos x sin x sin x sin x cos x sin x sin x sin x − = + + ⇔ − = ⇒ = 2 2 2 2 2 1 1 2 2 2 19/(2 2) (2 2) (2 2) (1 2 / 2) ( 2 0);20/ 2 2 ( 2) / 2 x x sin x cos x cos x cos x x x cos x x x − − + − + + − = + = − = − 4/ Một số dạng khác: 2 2 2 2 2 1 3 2 6 5 2 3 7 3 2 6 5 2 1 1/ 4 4 4 1 (4 1)(4 1) 0;2/( 2 1) 1 x x x x x x x x x x x x x x − − + + + + + − + + + + + = + ⇔ − − = − + ≥ 2 2 1 1 1 2 1 1 1 2 2 5 2 3/5.3 7.3 1 6.3 9 0 5.3 7.3 3 1 0;4/( ) 1 x x x x x x x x x x x − − + − − + − + − + − + = ⇔ − + − = − = 2 1 2 2 2 5/ 4 .3 4.3 1 0 4.3 4.3 1 (2.3 1) 0(*) x x x x x x+ − + ≤ ⇒ − + = − ≤ ⇒ BPT vô nghiệm vì x = 0 KTM (*). 2 2 2 3 2 3 4 1 ( 1) 2 1 2 1 2 1 1 6/ 4 2 2 1;7 / .2 2 .2 2 ;8/ .3 (3 2 ) 2(2 3 ) x x x x x x x x x x x x x x x x x − + − + + − + + − − − + = + + = + + − = − 2 3 2 2 2 1 2 9/ .3 3 .(12 7 ) 8 19 12;10/ 4 8 2 4 ( ).2 .2 . 2 x x x x x x x x x x x x x x x + + − = − + − + + − > + − + − 2 2 2 2 2 2 1 2 1 11/ 2 5 3 2 2 .3 . 2 5 3 4 .3 ;12/( 1/ 2) ( 1/ 2) x x x x x x x x x x x x x x + + − − − + > − − + + ≤ + 2 2 2 2 10 10 2 11 20 13/( 4 ) (4 ) ( 10; 1;4);14/( 2) ( 2) ( 1;2;3;4;5) x x x x x x x x x x x x − − + − − = − = ± − − = − = 2 3 2 3 1 2 2 2 2 2 1 5 15/1/(3 1) 1/(1 3 );16/( 1) 1 ;17/( 1) ( 1) x x x x x x x x x x x x x x + − + + + + − ≥ − − > − − + > − + 2 5 1 5 5 7 7 / 5 5 18/ 1 1( ; 1);19/3 ;20/ 5( );21/7 5 ( 7) x x n n x x n x n x x n n cosx x x t x log log + − + + − = = + = = = = ⇒ = 2 1 2 1 22/7 7 ( 0) 1 x x x nx n n x + + + + − + ≤ > ⇒ ≤ II. Phương trình, bất phương trình lôgarít: 1/ Đưa về 1 cơ số: 5 25 0,2 2 3 4 20 1/ 3;2/ 0,5 (5 4) 1 2 0,18;3/log x log x log lg x lg x lg log x log x log x log x+ = − + + = + + + = 2 5 1/ 5 5 1/ 25 4/ ( 6) 0,5 (2 3) 2 25;5/ ( 1) 5 ( 2) 2 ( 2)lg x lg x lg log x log log x log x+ − − = − + + = + − − 3 2 3 3 2 3 3 2 1 1 1 4 4 4 3 1 3 3 6/( ). 1; ;7/ ( 2) 3 (4 ) ( 6) 2 8 2 3 x log x log log log x x log x log x log x x   − = + = + − = − + +  ÷   2 DOÃN XUÂN HUY-THPT Ân Thi-Hưng Yên 2 2 0,5 0,25 2 0,5 2 8/ ( 3) 5 2 ( 1) ( 1)( 2)9/ (1 / 2) 2 / 4 0( 1)log x log log x log x log x log x+ + = − − + − + − = − [ ] 2 2 2 0,5 2 4 10/ 2 ( 1 ) ( 1 ) 3;11/ /(2 ) 0log x x log x x log tanx log cosx cosx sinx+ + + + − = + + = [ ] 2 5 5 5 5 25 0,2 12/ ( 6) ( 2);13/ 3;14/ ( 2 3) ( 3)/( 1)log x log x log x log x log x log lg x x lg x x= + − + + = + − + + − 2 2 2 2 3 6 0;15/ 0,5. (5 4) 1 2 0,18;16/ ( 1). ( 1) ( 1)lg x lg x lg log x x log x x log x x= − + + = + − − + − = − − 2 1/ 5 5 2 2 17/ ( 6 8) 2 ( 4) 0;18/ ( 3) 1 ( 1)log x x log x log x log x− + + − < + ≥ + − 2 / 3 3 8 1/ 8 0,5 3 3 2 19/ 2 ( 2) ( 3) 2/3;20/ 1( (1 3) 1 0 3 ) log log x log x log x log x log x log x− + − > + > ⇔ − > ⇒ < < 2 2 3 5 2 3 5 5 2 (2 ) 20/ . . ;21/ 3 4. 5 1;22/ ( 2). 2 2 0 x x log x log x log x log x log x log x log x log log x log − + + = + > + − ≥ ( 3) 0,25 2 2 3 3 2 23/ 6 2 (4 ) / ( 3) 1( 3); . 2 . 3 0(0 6 /6; 1) x log log x log x x log x log x log x log x x x +   + − + = = + ≥ < < ≥   2/ Đặt ẩn phụ: 0,04 0,2 16 2 1/1/(4 lg ) 2/(2 lg ) 1;2/ 1 3 1;3/3 16 4 2 x x x log x log x log log x log x− + + = + + + = − = 2 3 1 2 2 2 1/ 2 4/ 16 64 3;5/lg(lg ) lg(lg 2) 0;6/ (4 4). (4 1) 1/8 x x x x log log x x log log log + + = + − = + + = 2 2 (3 2 ) (3 ) 4 2 2 4 7/ (2 9 9) (4 12 9) 4 0;8/ ( ) ( ) 2( 4 1) t x x log x x log x x log log x log log x x t − − − + + − + − = + = = ⇒ = 2 4 4 3 2 2 2 2 2 25 9/ (2/ ). 1( ( 1)( 2 2 1) 0);10/ (125 ). 1(5 &1/ 625) x x log x log x log x t t t t t log x log x+ = ⇔ − + + + + = = 2 3 3 3 2 2 5 11/ 3 3 1/ 2;12/ (4 1) (2 6) ;13/ (5 ). 2 x x x x x log log x log log x log log x log x log x + + = + + + = − + = − 2 2 2 2 2 8 2 1 1 2 1 14/ 4 2 1 2 4 4 1: 3/ 2 (2 1) 2 1 2 1 2 x x x log log log t t t log x x x x   + −     + − = + ⇔ + = − − = ⇒ =  ÷  ÷  ÷ + − +       2 sin /16 / 64 5 5 15/ . 1/ 4;16/ 2. 2 2;17/ (5/ ) 1 xcosx sinxcosx x x x x log sinx log cosx log log log log x log x= = + = 1 2 2 18/ 10 6 0;19/lg(6.5 25.20 ) lg25;20/ 2(lg2 1) lg(5 1) lg(5 5) x x x x log x log x x − + + = + = + − + + = + 1/ 3 2 2 /16 2 21/ 5/ 2 3;22/ 2. 2. 4 1;23/ 2. 2 1/( 6) x x x x x log x log log log log x log log log x+ ≥ > > − 2 2 4 3 3 3 1/ 2 2 16 24/ 4 9 2 3;25/ 4 2(4 )log x log x log x log x log x log x− + ≥ − + < − 1 2 2 2 2 1/ 2 2 1/ 2 4 26/ (2 1). (2 2) 2;27/ 3 5( 3) x x log log log x log x log x + − − > − + − > − 2 2 2 2 2 2 2 6 4 63 5 28/ (2 ) (2 );29/ 2 2 (1/5);30/ 4 2.3 ( 6 ) log x log log x log log x x x x log x log x log x log x x≤ − ≥ − = = 3/ Phương pháp mũ hóa, lôgarít hóa: 2 3 5 11 1/4 4 5 5 2 3( 1) (lg 5) /3 5 lg 6 lg lg 3 1 1 2 ;2/ 10 ;3/ .5 11 ;4/ 2/ ( 1 1) ( 1 1) x log log log x log x x x x x x x x x x x − − − + + + + − −   = = = = + − − + +   2 2 2 1/ 3 4 9 2 (3 ) 5/ log ( 5) 0;6/ (3 9) 1;7/ ( 5 6) 1;8/ (3 ) 1 x x x x x log x log log log x x log x −     − > − < − + < − >     3 DOÃN XUÂN HUY-THPT Ân Thi-Hưng Yên 23 2 6 62 2 2 2 2 4 lg 3lg 1 2 9/ 1/ ;10/ 32;11/ 1000;12/ 6 12;13/ (4 6) 1 log x log x log x log x log x x x x x x x x x x log log − − + − +   > < > + ≤ − ≤   1/ 2 3 2 1/ 3 1/ 2 3 6 2 3 1 2 2 1 3 2 14/ 0;15/ ;16/ 0;17 / 1 1 2 2 2 2 x x x x x x x log log log log log log log log log x x x x x + + + − − + ≥ ≤ > > − − + + + [ ] 2 3 2 3 4 2 2 3 3 3/ 2 3 18/ ( ) 0;19/ ( 2 3 ( 2)) t t log log log x log log x log log x t x t log log= = = ⇒ = = ⇒ = 2 3 3 2 3 3 2 3 3 3 2 3 3 20/ ( / ) 2log log x log log x log log x log log x log log x log x log log x+ = ⇔ = = ⇒ = 2 3 3 2 3 log log 2 3 2 3 3 2 3 2 21/ 3 2 (2 3) 1 t x x log log log log log log x log log x log log t x≥ ⇔ − ≥ − ⇔ ≥ ⇒ < ≤ 2log 3 2 3 3 2 3 log    ÷   2 2 2 3 4 4 3 2 3 2 3 4 3 3 22/ ( 4) ( ) (2 )( 1)log log log x log log log x x log log x log log x log t log t t= > ⇔ = ⇔ = > 3 3 3 3 1 4 2 48 1 48log log log t log x∆ = + = ⇒ = + ⇒ = 1 48 3 3 4 log+ ; 2 23/ ( 9 1) 1 x log x x− − − ≥ 2 2 4 0,5 1,5 2 0,25 2 24/ 2. 2 ;25/ ( 3) 1 ( 4) ( / 3) 0 x log x log x x log x log x x log log x x −     ≥ + + ≥ ⇔ − + ≥     4/ Sử dụng tính đơn điệu của hàm số: 2 3 5 1/ lg( 6) 4 lg( 2) lg( 3) 4 4;2/ ( 1) (2 1) 2 2x x x x x x x log x log x x+ − − = + + ⇔ + − = ⇒ = + + + = ⇒ = 2 3 3 3 3/( 2) ( 1) 4( 1) ( 1) 16 0( ( 1) 4;4/( 2) 80/81;2)x log x x log x log x x x+ + + + + − = + = − + ⇒ = − 2 2 2 9 32 2 3 2 4/ (1 ) ( 1 3 2 2);5/ .3 ( 9 12 3 ) t log log x logt t t t log x log x t t x x x log x t+ = = ⇒ + = ⇒ = = − = ⇒ = − 5 ( 3) 6 3 3 2 5 2 6/3 (1 ) 2 ( 2 1 8 4 9 2);7/ 2 ( 3) log x t t t t log x x log x x t x log x log x + + + = = ⇒ + + = ⇒ = = ⇔ + = ( ) 6 6 6 2 6 2 3 5 2;8/ 3 3 2 6 3 2 1 1/ 6 log x log x log x t t t t t t x log x log x x t x = ⇒ + = ⇒ = + = ⇔ + = ⇔ + = ⇒ = − ⇒ = 2 2 2 1 3 2 2 3 9/3 1;10/ 2 2 8/ (4 4 4)( , 1/ 2) log x x x x log x x VP VT x + − = − + = − + ≤ = 7 3 7 3 11/ ( 2)( ( 7 2) 3 7 2 1 (2) ( ) t t t log x log x log x t t log f f t< + = ⇒ < + ⇔ < + ⇔ = < = 7 ( 7 /3) 2.(1/3) 2 49 0) t t t log x x+ ⇒ > = ⇒ > > 2 2 2 2 2 2 3 2 1/3 3 3 12/ 4 ( 2 3) 2 (2 2 2) 0 2 ( 2 3) x x x x x log x x log x log x x − − − + − + − + + − + = ⇔ − + = 2 2 2 2 3 2 (2 2 2) 2 3 2 2 2 3 x log x x x x x − + − + ⇔ − + = − + ⇒ = − 2 2 2 2 2 3 2 3 13/ ( 5 5 1) ( 5 7) 2( 5 5 ( ) ( 1) ( 2) 2log x x log x x t x x f t log t log t− + + + − + ≤ = − + ⇒ = + + + ≤ (1) 0 1 1 (5 5)/ 2 (5 5)/ 2 4f t x x= ⇒ ≤ ≤ ⇒ ≤ ≤ − ∧ + ≤ ≤ 2 2 2 2 2 2 14/ 2 (4 2) 1 2 ( 2) 2 . 1 2 x x log x x log x VT VP x − − −   − − ≥ ⇔ − − ≥ ≤ ≤ ⇒ =   3 2 2 3 4 5 15/ 2 cot ( 1);16/ ( 1) ( 2) ( 3)log x log cosx t t log x log x log x log x= = ⇒ = − + + = + + + 2 4 ( ) ( 2) '( ) 1/ ln 2 1/( 2)ln 4 0 0f x log x log x f x x x x= − + ⇒ = − + > ∀ > ⇒ f(x) đồng biến khi x > 0. Tương tự 3 5 ( ) ( 1) ( 3)g x log x log x= + − + cũng đồng biến khi x > 0. Suy ra pt có nghiệm dn x = 2. 4 DOÃN XUÂN HUY-THPT Ân Thi-Hưng Yên [ ] 2 1/ 2 1/ 2 2 2 16/( 1) (2 5) 6 0 ( 2) ( 1) 3 0 0 2 4x log x x log x log x x log x x x+ + + + ≥ ⇔ − + − ≥ ⇒ < ≤ ∧ ≥ 2 2 2 3 2 5 11 5 11 2 ( 4 11) ( 4 11) 17/ 0( 4 11 0; ( ) 2 3 ; 2 5 3 log x x log x x t x x f t log t log t x x − − − − − ≥ = − − > = − − − [ ] '( ) ln(121/125) / ln5.ln11 0 0;0 (1) 2 15 6f t t t f x x= < ∀ > = ⇒ < − ∧ ≥ 2 2 2 (2 3) 2 2 3 18/ ( 2 2) ( 2 3); 2 3 2; 2 3 0log x x log x x a t x x + + − − = − − = + < = − − > 2 2 2 ( 1) 1 (2 ) ( / 2) (1/ 2 ) 1 2 1 11 4 3 u u u u a a log t log t u a a a a u x⇒ + = = ⇒ + = ⇔ + = ⇒ = ⇒ = ± + 2 2 2 1/ 3 1/ 3 2 19/( 1) 2( 3) 8 0;20/ 2 8 (2 1)/( 1)x log x x log x x x log x x   + + + + ≤ − = + − ⇔   2 2 2 2 2 2( 1) ( 1) (2 1) 2(2 1) ( 1) 2 1 0;4x log x log x x x x x− + − = + + + ⇔ − = + ⇒ = 5/ Một số Phương trình, bất phương trình khác: [ ] 2 2 2 ( 6) 2 1/ 3 1/ 3 1/1/ 2 3 1 1/ ( 1) (0;1/ 2) (1;3/ 2) (5; ) ;2/(2 3.2 ) 1( 1) log x log x x x log x x log x a − + − − + > + ∪ ∪ +∞ + > > 3/ ( 1) lg1,5(0 1 0 ; 1 1 ) x log x x VT VP x VT VP+ = < < ⇒ < < > ⇒ > > 2 2 2 2 2 2 4/ ( 3 1) 2 0 0 ( 2)( 3) 1& 3 2 3 1 0log x x log x t t t t t x+ − − + ≤ ⇔ < − + − ≤ > ⇒ > > ⇒ > > 1 2 2 3 3 5/ (3.2 1) / 1( 1 (2/3) 0);6/( 1)/ (9 3 ) 3 1( log 9 3 0) x x log x x log x x log MS −     − ≥ ≥ ∧ < < − − − ≤ < − <     [ ] 5 5 3 3 4 4 7/ ( /3) (2 )/ ((0; 5 /5) (1;3));8/1/ ( 1) /( 2) 1/ ( 3) x log x log x log x log x log x log x x log x+ < − ∪ + + < + III. Hệ phương trình, bất phương trình mũ và lôgarít: 3 1 2 3 2 3 2 1 2 2 2 3.2 3 2 77 2 5 4 2 2 12 1/ ;2/ ;3/ ;4/ ; 5 3 2 7 4 2 (2 2) 3 1 1 x y y x x y x x y x y x x x y y x y y x xy x + − + +  + =   − = = −  + =        + = − = + = + + + = +        2 3 2 3 5 1 2 1 4 2 4 5( ) 26 4 3.4 2(1) 3 5 (1) 5/ ;6/ ;7 / 3 2 3(2) 64 4 1 ( 3) 8(2) x x log x y y y x y log y log x x y log xy y y y − − − + − − − −  + =   + ≤ =     + ≥ − = − − + + ≤     8 8 12 ( ) 2 2 2 x+y 3 3 3 3 4 4 (1 2 2) 1 3 4 8/ ;9/ ;10/ ;11/ 12 1 2 3 x y log y log x x y xy log log xy xlog log y y log x x y x y xlog log x y log y log x log y y x x y + +  + =  + = + =  + =       + = + − = =   − =     2 2 1 2 2 2 2 2 2 2 2 3 2 ( )( 1) 0(1) .2 3 .2 2 12/ ;13/ ;14/ 2 .2 3 .8 1 1 /3 3 5 9 0(2) x y x y x y x y x y e e log x log y xy log x log x x y x y x y x x x − + + + +    − = − + − < + =       + = + = − + + >       5 DOÃN XUÂN HUY-THPT Ân Thi-Hưng Yên 1 1/ 4 4 2 2 2 3 9 3 ( ) (1/ ) 1 1 2 1 ( 1)lg2 lg(2 1) lg(7.2 12) 15/ ;16/ ;17/ ( 2) 2 25 3 (9 ) 3 x x x log y x log y x y x log x x y log x log y +  − − =  − + − = − + + < +     + > + = − =     Gợi ý một số bài: Bài 5: 2 2 3 3 2 (1) 3 5 1 3 0 3 (2): 4 1 ( 3) 8 x x y y y y y y − − − − ⇔ = ≥ ⇒ − − ≥ ⇒ ≤ − ⇒ − + − + + ≤ ( 3) 0 3 0 3 1;3y y y y x⇔ + ≤ ⇔ − ≤ ≤ ⇒ = − ⇒ = − Bài 6: 4 1 2 3 2 1 2 1 2 2 1 4 (2) 1 1 2 3;(1) 2 4 3.4 (3.4 1) 0 4 1/3 y log y y y x y y log − − − − − ⇒ + − ≥ − − ⇒ ≥ + ⇔ − ≤ ⇒ = 4 4 0,5 (4/3); 2 (9 3 /8)y log x log⇒ = ≥ − Bài 14: (1) có nghiệm ( 1; 4 ). Hàm số vế trái của (2) dương trên khoảng ( 1; 4 ) nên hệ có nghiệm là khoảng ( 1; 4 ). ------------------ // ------------------ 6