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<span class='text_page_counter'>(1)</span>DOÃN XUÂN HUY-THPT Ân Thi-Hưng Yên. PHƢƠNG TRÌNH, HỆ PHƢƠNG TRÌNH, BẤT PHƢƠNG TRÌNH MŨ VÀ LÔGARÍT I.Phương trình, bất phương trình mũ : 1/ Đƣa về cùng một cơ số hoặc hai cơ số: 1/ 2. x 2 x 8. 13 x. 4. ;2/ 3. x 1. 3. x 2. x 1 x 1. 3. x 3. 3. x 4. 750;3/ 5 .8 x. x 1 x. 500 (5.21/ x ) x3 1 x 3; log0,2 2. x 1 x (2; 1) (1; ) x 1 x 5/ 9 x 9 x1 9 x2 4 x 4 x1 4 x2 9 x.91 4 x.21 9/ 4 21/ 91 x log 9 / 4 (21/ 91) 4/( 5 2). x 1. ( 5 2). x 1 . 6/ 2 x .4 x 256;7 / 2 x.5x 0,01;8/ 2 x . 3x 216;9/(3 3 3 ) x (1/81)2 x3 ;10/ 2 x.3x1.5x2 12 2. 11/ 2. x 2 4. 5. x 2. ;12/8. x x2. 2 x. 36.3 ;13/1 5. x2 x. 1/ 2 x 1. 25;14/ 2. 1/(3 x 1). 2. ;15/( 10 3). x 3 x 1. ( 10 3). x 1 x 3. 2/ Đặt ẩn phụ: 1/(7 4 3) x 3(2 3) x 2 0(t 2 3/ t 2 0);2/(3 5) 2 x x (3 5) 2 x x 212 x x 0 2. 2. (t 1/ t 2 0);3/ 23 x 6.2 x 1/ 23( x1) 12/ 2 x 1(t 2 x 21 x );4/ 32 x 8.3x. ( chia 2 vế cho 32 x ); 5/ 4 x. x 2 2. 5.2 x1. x 2 2. 7 / 27 x 6.64 x 6.36 x 11.48 x ;8/ 22 x 2 x 2. 2. 2 x 1. 2. 2. 1. 9.2x. 2. x. x4. 0. 24 x1;9/( 5 2 6 ) x ( 5 2 6 ) x 10 2. 2. 9.9. 6 0;6/ 432cosx 7.41cosx 2 0 ;. 72 x 2.3x 2 x2 2t 4 1 x 1 x 10 / 6.(0,7) 7;11/ 1 1 ;12 / 3. x x x 100 3 2 t 1 3 3 13/ 9sin x 9cos x 10;14/ 22 x. x4. 2. 22 x2 0;15/ 22. x 3 x 6. 15.2. x 1 x. 12. x 3 5. 2x ;16/ 9 x 3x2 3x 9. 17 / 25x 10x 22 x1;18/ 4 x 2.6 x 3.9 x ;19/ 4.3x 9.2 x 5.6 x / 2;20/125 x 50x 23 x1 . 3/ Sử dụng tính đơn điệu của hàm số: 1/ 2x 1 3x / 2 ;2/ 2 x1 3x1 6 x 1;3/(2,5) x (0,4)1/ x 2,9;4/ 3. 1 Lop12.net. x 4. 2. 2 x 4. 13;5/ 2 x 6 x.
<span class='text_page_counter'>(2)</span> DOÃN XUÂN HUY-THPT Ân Thi-Hưng Yên 6/ 2x1 2 x x ( x 1)2 ;7 / 2 3 x x2 8 x 14;8/ 3x 6 x10 x2 6 x 6;9/ 3x 5x 6 x 2 10/ 32 x3 (3x 10).3x2 3 x 0;11/ 3.25x1 (3x 7).5x1 2 x 0;12/ x2 (3 2 x ) x 2 2 x1 0 2. 2. 32 x 3 2 x 13/ 3 3 2 2 6 2 x 6;14/ 2 3 5 2 3 5 ;15/ 0. 4x 2 2 2 y 16/ 4sin ( x ) 4cos ( x ) 8 x2 12 x 1/ 2(3/ 4);17 / 4sinx 21sinx cos( xy) 2 0( k ;0) x. x. x. x. x. 2 x 1. 2 x 1. 2x. x 1. x. . x 2. . 18/ 1 sin2 x.2cos 2 x 0,5.sin2 2 x cos 2 x sin2 x 2sin2 x 212 sin x 0 sin2 x 0;0,5 19/(2 2). sin2 x. (2 2). cos 2 x. (2 2). 2. (1 2 / 2). cos 2 x. cos 2 x. (cos2 x 0);20/ 2. 1 x 2 x2. 2. 12 x x2. ( x 2) / 2 x. 4/ Một số dạng khác: 1/ 4 x. 2. 3 x 2. 4x. 2. 6 x 5. 42 x. 2. 3 x 7. 1 (4 x. 2. 3 x 2. 1)(4 x. 2. 6 x 5. x 1. 1) 0;2/( x2 2 x 1) x1 1. 3/ 5.32 x1 7.3x1 1 6.3x 9 x1 0 5.32 x1 7.3x1 3x1 1 0;4/( x x2 ) 2 x. 2. 5 x 2. 1. 5/ 4x 1.32 x 4.3x 1 0 4.32 x 4.3x 1 (2.3x 1) 2 0(*) BPT vô nghiệm vì x = 0 KTM (*). 2. 6/ 4x. 2. x. 21 x 2( x1) 1;7 / x2 .2 x1 2 2. 2. x 3 2. x2.2. x 3 4. 2 x1;8/ x2.3x1 x(3x 2 x ) 2(2 x 3x1). 9/ x2 .3x 3x.(12 7 x) x3 8 x 2 19 x 12;10/ 4 x 8 2 x 2 4 ( x 2 x).2 x x.2 x1. 2 x 2 11/ 2 5 x 3x2 2 x 2 x.3x. 2 5 x 3x 2 4 x 2.3x ;12/( x 2 1/ 2) 2 x 13/( x2 4 x) x 15/1/(3. x 1. 18/ n x. 22/ 72 x. 2. 10. (4 x) x. 2. 10. ( x 10; 1;4);14/( x 2) x. 1) 1/(1 3 );16/( x 1). n1. x 1. x. 2. x2 2 x. 3. 2. 2 x. 2. x 1. ( x 2 1/ 2)1 x. ( x 2)11x20 ( x 1;2;3;4;5). x 1 ;17 /( x x 1) 2. 2. x 3 x 1. ( x x 1) 2. x 2 x 5. n 1 x 1( x n; n 1);19/ 3x cosx;20/ x x 5( x5 t );21/ 75 57 x log7 / 5 (log5 7) n. 7 2. x 1. 2. 5. x. x. nx n(n 0) x 1. II. Phương trình, bất phương trình lôgarít: 1/ Đƣa về 1 cơ số: 1/ log5 x log25 x log0,2 3;2/ 0,5lg (5x 4) lg x 1 2 lg 0,18;3/ log2 x log3 x log4 x log20 x 4/ lg ( x 6) 0,5lg (2 x 3) 2 lg 25;5/ log5 ( x 2 1) log1/ 5 5 log5 ( x 2) 2log1/ 25 ( x 2). 2 Lop12.net.
<span class='text_page_counter'>(3)</span> DOÃN XUÂN HUY-THPT Ân Thi-Hưng Yên 3 x3 1 3 3 2 3 3 6 /(log 2 x).log3 log3 log 2 x x 1; ;7 / log 1 ( x 2) 3 log 1 (4 x) log 1 ( x 6) x 8 2 3 2 4 4 4 2 8/ log2 ( x 3) log0,5 5 2log0,25 ( x 1) log2 ( x 1)( 2)9/ log0,5 (1 x / 2) log2 2 x / 4 0(1) 10/ 2log2 ( x 2 1 x) log0,5 ( x 2 1 x) 3;11/ log 2 tanx log 4 cosx /(2cosx sinx) 0. 12/ log5 x log5 ( x 6) log5 ( x 2);13/ log5 x log25 x log0,2 3;14/ lg ( x 2 2 x 3) lg ( x 3) /( x 1). 0;15/ 0,5.lg (5 x 4) lg x 1 2 lg 0,18;16/ log2 ( x x 2 1).log3 ( x x 2 1) log6 ( x x 2 1) 17 / log1/ 5 ( x2 6 x 8) 2log5 ( x 4) 0;18/ log2 ( x 3) 1 log2 ( x 1) 19/ 2log8 ( x 2) log1/ 8 ( x 3) 2/ 3;20/ log0,5 x log3 x 1( log3 x(1 log2 3) 1 0 x 3log2 / 3 3 ) 20/ log2 x log3 x log5 x log 2 x.log3 x.log5 x;21/ log5 3x 4.log x 5 1;22/ log 2 ( x 2 2).log (2 x ) 2 2 0 23/ log( x3) 6 2log0,25 (4 x) / log2 ( x 3) 1( x 3); log2 x.log3 2 x log3 x.log2 3x 0(0 x 6 / 6; x 1). 2/ Đặt ẩn phụ: 1/1/(4 lg x) 2/(2 lg x) 1;2/ log0,04 x 1 log0,2 x 3 1;3/ 3log x16 4log16 x 2log2 x. 4/ log x2 16 log2 x 64 3;5/ lg(lg x) lg(lg x3 2) 0;6/ log 2 (4 x1 4).log 2 (4 x 1) log1/. 2. 1/8. 7 / log(32 x ) (2 x2 9 x 9) log(3 x ) (4 x 2 12 x 9) 4 0;8/ log4 (log2 x) log2 (log4 x) 2( x 4t t 1) 2 9/ log2 x (2/ x).log22 x log24 x 1( (t 1)(t 4 2t 3 t 2 2t 1) 0);10/ log x (125x).log 25 x 1(5 &1/ 625). 11/ log x 3 log3 x log x 3 log3 x 1/ 2;12/ log 2 (4 x 1) log 2 (22 x3 6) x;13/ log x (5 x).log5 x 2 8x 1 2x 1 2x 1 14 / 4 2log 2 1 log 2 log 4 t 4 t 1: t log x 3/ 2 2 2 2 2 2x 1 2 2x 1 (2 x 1) 2 15/ logsin xcosx sinx.log sinxcosxcosx 1/ 4;16/ log x 2.log x /16 2 log x / 64 2;17 / log5 x (5/ x) log5 x 1 18/ log2 x 10log2 x 6 0;19/ lg(6.5x 25.20 x ) x lg 25;20/ 2(lg 2 1) lg(5 x 1) lg(51 21/ log1/ 3 x 5/ 2 log x 3;22/ log x 2.log2 x 2.log2 4 x 1;23/ log x 2.log x /16 2 1/(log2 x 6). x. 5). 24/ log32 x 4log3 x 9 2log3 x 3;25/ log1/2 2 x 4log 2 x 2(4 log16 x 4 ) 26/ log2 (2 x 1).log1/ 2 (2 x1 2) 2;27 / log22 x log1/ 2 x 2 3 5(log4 x 2 3). 28/ log x (2 x) log x (2 x3 );29/ 2log5 x 2 log x (1/ 5);30/ 4log2 2 x xlog2 6 2.3log2 4 x ( xlog2 6 6log2 x ) 2. 3 Lop12.net.
<span class='text_page_counter'>(4)</span> DOÃN XUÂN HUY-THPT Ân Thi-Hưng Yên 3/ Phƣơng pháp mũ hóa, lôgarít hóa: log5 5. xlog4 x2 23(log4 x1) ;2/ x(lg x5) / 3 105lg x ;3/ x6 .5log1/ x 5 11. 11. ;4/ xlg. 2. x lg x3 3. 2/ ( x 1 1) 1 ( x 1 1) 1. 5/ log1/ 3 log4 ( x 2 5) 0;6/ log x log9 (3x 9) 1;7 / log2 x ( x 2 5 x 6) 1;8/ log(3 x x2 ) (3 x) 1. 9/ x2log2 2 xlog2 x 1/ x;10/ xlog2 x4 32;11/ xlg 3. 14 / log1/ 2log3. 2. x 3lg x 1. 1000;12/ 6log6 x xlog6 x 12;13/ log x log 2 (4 x 6) 1 2. x 1 x2 x2 x 1 3x 2 0;15/ log 2log1/ 3 log1/ 2log3 ;16 / log x6 log 2 0;17 / log x 1 x 1 x2 x2 x 2 x 2 3. 18/ log2 log3 (log4 x) 0;19/ log2log2 x log3log3 x( t x 22 33 t log3/ 2 (log3 2)) t. t. 2log3log3 2. 20/ log2log3 x log3log2 x log3log3 x log2log3 x log3 (log3 x / log 2 x) log3log3 2 x 3. 2 log 2 3 log3 2 3 . 21/ log2log x 3 log3log x 2 log2log3 x log3log2 x log3 (2t log 2 3) t 1 x 3. 22/ log2log3log4 x log4log3log2 x( x 4) log3log2 x (log3log 4 x)2 log32t log3 (2t )(t 1) 1 log3 48. 3. 1 4log3 2 log3 48 log3t 1 log3 48 x 4 24/ 2.x0,5log2 x 21,5log2 x ;25/ log2 x log0,25 ( x 3) . x 4. ; 23/ log x ( 9 x 2 x 1) 1. 1 ( x 4)log2 log ( x / x 3) 0. 4/ Sử dụng tính đơn điệu của hàm số: 1/ x lg( x2 x 6) 4 lg( x 2) x lg( x 3) 4 x 4;2/ log3 ( x 1) log5 (2 x 1) 2 x 2 3/( x 2)log32 ( x 1) 4( x 1)log3 ( x 1) 16 0(log3 ( x 1) 4;4/( x 2) x 80/81;2) t. 4/ log2 (1 x ) log3 x( t 1 3 2t t 2);5/ xlog2 9 x 2.3log2 x xlog2 3 (log 2 x t 9t 12t 3t ) 6/ 3log3 (1 x 3 x ) 2log2 x ( x 26t 1 8t 4t 9t t 2);7 / 2log5 ( x3) x log5 ( x 3) log 2 x. . . t 2t 3 5t x 2;8/ log2 x 3log6 x log6 x x 3log6 x 2log6 x 6t 3t 2t t 1 x 1/ 6 9/ 3log2 x x2 1;10/ 22 x1 232 x 8/ log3 (4 x2 4 x 4)(VP VT , x 1/ 2) t. t. 11/ log7 x log3 ( x 2)(log7 x t t log3 ( 7 2) 3t 7 2 1 f (2) f (t ) ( 7 / 3)t 2.(1/ 3)t 2 t log7 x 49 x 0). 12/ 4. 2. x 2. 2 x 2 2. log 3 ( x 2 2 x 3) 2 x. 2. 2 x. log1/ 3 (2 x 2 2) 0 2 x. log3 (2 x 2 2) x2 2 x 3 2 x 2 2 x 3. 4 Lop12.net. 2. 2 x 3. log3 ( x 2 2 x 3) .
<span class='text_page_counter'>(5)</span> DOÃN XUÂN HUY-THPT Ân Thi-Hưng Yên. 13/ log2 ( x 2 5x 5 1) log3 ( x 2 5x 7) 2(t x 2 5 x 5 f (t ) log 2 (t 1) log3 (t 2 2) 2 f (1) 0 t 1 1 x (5 5) / 2 (5 5) / 2 x 4. 14/ 2. x 2. x 2 log2 (4 x x 2 2) 1 log2 2 ( x 2)2 2 .VT 1 VP x 2. 15/ 2log3 cot x log2cosx( t t 1);16/ log2 x log3 ( x 1) log4 ( x 2) log5 ( x 3) f ( x) log2 x log4 ( x 2) f '( x) 1/ x ln 2 1/( x 2)ln 4 0x 0 f(x) đồng biến khi x > 0. Tương tự g ( x) log3 ( x 1) log5 ( x 3) cũng đồng biến khi x > 0. Suy ra pt có nghiệm dn x = 2.. 16/( x 1)log1/2 2 x (2 x 5)log1/ 2 x 6 0 (log2 x 2) ( x 1)log2 x 3 0 0 x 2 x 4. log5 ( x 2 4 x 11)2 log11 ( x 2 4 x 11)3 17 / 0(t x 2 4 x 11 0; f (t ) 2log5t 3log11t; 2 2 5 x 3x f '(t ) ln(121/125) / t ln5.ln11 0t 0;0 f (1) x 2 15 x 6 18/ log. 2 2 3. ( x 2 2 x 2) log(2. 3). ( x 2 2 x 3); a 2 3 2; t x 2 2 x 3 0. log2 a (t 1) log a2 t u a 2u 1 (2a)u (a / 2)u (1/ 2a)u 1 u 2 x 1 11 4 3. 19/( x 1)log1/2 3 x 2( x 3)log1/ 3 x 8 0;20/ 2 x 2 8x log 2 (2 x 1) /( x 1) 2 2( x 1)2 log2 ( x 1)2 log2 (2 x 1) 2(2 x 1) ( x 1)2 2 x 1 x 0;4. 5/ Một số Phƣơng trình, bất phƣơng trình khác: 1/1/ log1/ 3 2 x 2 3x 1 1/ log1/ 3 ( x 1) (0;1/ 2) (1;3/ 2) (5; ) ;2/(2 x 3.2 x ) 2log2 xlog2 ( x6) 1(a 1) 3/ log x ( x 1) lg1,5(0 x 1 VT 0 VP; x 1 VT 1 VP). 4/ log2 ( x2 3 x2 1) 2log2 x 0 0 (t t 2 2)(t 2 3) 1& t 3 2 t 3 1 x 0 5/ log2 (3.2 x1 1) / x 1( x 1 log2 (2/ 3) x 0);6/( x 1) / log3 (9 3x ) 3 1( MS log 3 9 3 0). 7 / log5 x log x ( x / 3) log5 x(2 log3 x) / log3 x((0; 5 / 5) (1;3));8/1/ log 4 ( x 1) /( x 2) 1/ log 4 ( x 3). III. Hệ phương trình, bất phương trình mũ và lôgarít: 23 x1 2 y 2 3.2 y 3 x 32 x 2 y 77 2 x 2 y 12 23 x 5 y 2 4 y 1/ x ;2 / ;3/ x ;4 / ; y x 1 x 2 x y 5 3 2 7 4 2 y (2 2) 3x 1 xy x 1. 5 Lop12.net.
<span class='text_page_counter'>(6)</span> DOÃN XUÂN HUY-THPT Ân Thi-Hưng Yên x 2 2 x 3 log3 5 4 x y 1 3.42 y 1 2(1) 5(log x y log y x) 26 5 y 4 (1) 3 5/ ;6 / ;7 / 2 x 3 y 2 log 4 3(2) xy 64 4 y y 1 ( y 3) 8(2) x y 12 (1 2log xy 2)log( x y ) xy 1 xlog8 y y log8 x 4 xlog 2 3 log 2 y y log 2 x x y 8/ x+y ;9 / ;10 / ;11/ 3 xlog 12 log x y log y y x 3 3 3 log 4 x log 4 y 1 x y 2 3 e x e y (log 2 y log 2 x)( xy 1) log 22 x log 2 x 2 0(1) x.2 x y 1 3 y.22 x y 2 12 / 2 ;13/ ;14 / 3 2 x y x y 2 2 2 x .2 3 y .8 1 x y 1 x / 3 3x 5 x 9 0(2) . ( x 1)lg 2 lg(2 x1 1) lg(7.2 x 12) log1/ 4 ( y x) log 4 (1/ y) 1 x 1 2 y 1 15/ ;16 / ;17 / 2 2 2 3 log x ( x 2) 2 x y 25 3log9 (9 x ) log3 y 3. Gợi ý một số bài: x 2 2 x 3. Bài 5: (1) 3 5 y 3 1 y 3 0 y 3 (2) : 4 y y 1 ( y 3) 2 8 y( y 3) 0 3 y 0 y 3 x 1;3 Bài 6: (2) x y 1 1 2 y log4 3;(1) 2 412 ylog4 3 3.42 y1 (3.42 y1 1)2 0 42 y1 1/ 3 y 0,5log4 (4/ 3); x 2 log4 (9 3 /8). 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 Lop12.net.
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