alimov-10-gdz (Алгебра - 10-11 класс - Алимов), страница 13

1) 8log 2 5 = 23log 2 5 = (2log 2 5 )3 = 53 = 125 ;2) 9log 3 12 = 32log 3 12 = (3log 3 12 ) 2 = 122 = 144 ;3) 16log 4 7 = 42log 4 7 = (4log 4 7 ) 2 = 7 2 = 49 ;4) 0,125log 0,5 13log0,5 1= 0,5= (0,5log 0,5 1 3) = 13 = 1 .277. 1) log6 x = 3 ⋅ 1; log 6 x = 3 log 6 6; log 6 x = log 6 6 3 ; x = 6 3 = 216 ;2) log 5 x = 4 ⋅1; log 5 x = 4 log 5 5; log 5 x = log 5 5 4 ; x = 5 4 = 625 ;3) log 2 (5 − x ) = 3 ⋅ 1; log 2 (5 − x ) = 3 log 2 2; log 2 (5 − x ) = log 2 2 3 ;5 − x = 2 3 ; 5 − x = 8; x = −3 ;4) log3 ( x + 2) = 3 ⋅ 1; log 3 ( x + 2) = 3 log 3 3; log 3 ( x + 2) = log 3 3 3 ;x + 2 = 3 3 ; x + 2 = 27; x = 25 ;15) log 1 (0,5 + x) = −1 ⋅ 1; log 1 (0,5 + x) = −1 ⋅ log 1 ;666 6−11log 1 (0,5 + x) = log 1   ; 0,5 + x = 6; x = 5,5 .66 6278.

1) log 1 (4 − x) существует при 4 − x > 0; x < 4 ;22) log0, 2 (7 − x ) существует при 7 − x > 0; x < 7 ;111> 0; 1 > 2x; x < ;существует при1 − 2x1 − 2x2551> 0; 2 x − 1 > 0; x < ;4) log 8существует при2x − 12x − 123) log686www.5balls.ru5) log 1 (− x 2 ) существует при − x 2 > 0 — не имеет действительных ре4шений, значит log 1 (− x 2 ) — не существует;436) log0,7 (−2x ) существует при − 2x 3 > 0; x < 0 .1279. 1) log 2 4 2 = log 2 2 4 =12) log 311⋅ log 2 2 = ;44= log 3 3 −1,5 = −1,5 ⋅ log 3 3 = −1,5 ;3 353) log 0,51 1 2 5= log 0,5   = ⋅ log 0,5 0,5 = 2,5 ;232 231−2+722= log 7 7 3 = −1 ⋅ log 7 7 = −1 .49334) log 7280.

1) 92log3 5 = 34log3 5 = (3log3 5 )4 = 54 = 625 ;1 1 22)  913)  4log 43= 3−1⋅log3 4 = (3log3 4 )−1 = 4−1 =−5log 2 3−4 log 1 54) 273= 2( −2)⋅( −5) log 2 3 = (2log 2 3 )10 = 310 = 59049 ;1= 35) 103−log10 5 =7( −3)( −4) log 1 510310log10 51+ 2 log 1 316)  71;43=12 1 log 1 5 3 =3= 512 ;1000= 200 ;52log 31  1  1 12= ⋅    7  = ⋅ 32 = 1 .7  7 77281. 1) log 2 (log3 81) = log2 (log3 34 ) = log2 (4(log3 3)) = log2 22 = 2 ⋅ log2 2 = 2 ;2) log 3 (log 2 8) = log 3 (log 2 2 3 ) = log 3 (3 ⋅ log 2 2) = log 3 3 = 1 ;3) 2 log 27 (log10 1000) = 2 log 27 (log10 10 3 ) = 2 log 27 (3 log10 10) =122= 2log 27 3 = 2log 27 27 3 = log 27 27 = ;3311134) log 9 (log 2 8) = log 9 (log 2 2 ) = log 9 (3 log 2 2) =333111111= log9 3 = log9 9 2 = ⋅ log9 9 = ;333 2687www.5balls.ru−115) 3log 2 (log 4 16) + log 1 2 = 3log 2 (log 4 42 ) + log 1   =2221= 3log 2 (2log 4 4) − log 1 = 3log 2 2 − 1 = 3 − 1 = 2 .2 2282.

1) log x 27 = 3; log x 27 = 3 log x x; logx27=logxx3; x3=27; x3=33; x=3;1 1111= ; x=7;= −1; log x = −1 ⋅ log x x; log x = log x x −1 ;7777 x13) log x 5 = −4; log x 5 = −4 log x x; log x 5 = log x x −4 ; 5 =;x42) log x1 1 8x =; x =  .5514283. 1) log 6 (49 − x 2 ) — существует при 49 − x 2 > 0; −7 < x < 7 ;2) log7 ( x 2 + x − 6) — существует при x 2 + x − 6 > 0; x < −3 и x > 2 ;3) log 1 (x 2 + 2x + 7) — существует при х2 + 2х + 7 > 0, т.е. при любом x .5284.

1) log 3 (1 − x 3 ) — существует при 1 − x 3 > 0; x 3 < 1; x < 1 ;2) log 2 ( x 3 + 8) — существует при x 3 + 8 > 0; x 3 > −8; x > −2 ;3) log 1 (x 3 + x 2 − 6x) — существует при x 3 + x 2 − 6x > 0;42x ( x + x − 6) > 0; −3 < x < 0 и x > 2 ;4) log 1 (x 3 + x 2 − 2x) — существует при x 3 + x 2 − 2 x > 0;3x ( x 2 + x − 2) > 0; −2 < x < 0 и x > 1 .285.

1) 2 x = 5; x = log 2 5 ;2) 1,2 x = 4; x = log1, 2 4 ;1(log 4 5 − 3) ;21= 2; 1 − 2x = log7 2; x = (1 − log 7 2) .23) 42 x +3 = 5; 2 x + 3 = log 4 5; x =4) 71− 2 x286. 1) 7 2 x + 7 x − 12 = 0; 7 x = t; t 2 + t − 12 = 0; t = −4 — постороннийкорень, t = 3; 7 x = 3; x = log 7 3 ;2) 9x – 3x – 12 = 0; 32x – 3x – 12 = 0; 3x = t; t2 – t – 12 = 0; t = – 3 — посторонний корень, t = 4; 3x = 4; x = log34.;13) 8 x +1 − 82 x −1 = 30; 8 x = t; t 2 − 8t + 30 = 0; t 2 − 64 t + 240 = 0; t = 4 ;888www.5balls.rut1 = 3; 8x = 4; 2 3x = 2 2 ; 3x = 2; x1 =x2;3t 2 = 60; 8 x = 60; x 2 = log8 60 ;xxx−1x4)  1  − 5 1  + 6 = 0;  1  = t; t 2 − 5t + 6 = 0; t1=3  1  = 3;  1  =  1  ;933 3 3 3x1 = −1 ; t 2 = 2; x 2 = log 1 2 .3xxxx287. 1) (3 + 2 )(3 + 3 ⋅ 2 ) = 8 ⋅ 6x ; 32 x + 3 ⋅ 6x + 6x + 3 ⋅ 22 x − 8 ⋅ 6 x = 0;32 x − 4 ⋅ 6x + 3 ⋅ 22 x = 0;  3 22xxx233−   + 3 = 0;   = t; t − 4t + 3 = 0; t1 = 3;22xx33  = 3; x1 = log 3 3; t 2 = 1;   = 1; x = log 3 1; x 2 = 022222) (3 ⋅ 5x + 2,5 ⋅ 3x )(2 ⋅ 3x − 2 ⋅ 5x ) = 8 ⋅ 15x ;6 ⋅ 15x − 6 ⋅ 52 x + 5 ⋅ 32 x − 5 ⋅ 15x − 8 ⋅ 15x = 0; 5 ⋅ 32 x − 7 ⋅ 15x − 6 ⋅ 52 x = 0;35⋅ 52xxx23 3− 7  − 6 = 0; t =   ; 5t − 7 t − 6 = 0; t = −0,6 — посторон55xний корень, t = 2;  3  = 2; log 3 2 = x .55x > 0288.

1) log x (2x − 1) существует при  x ≠ 1;2x − 1 > 0x > 0 1x ≠ 1 ; 2 < x < 1 и x > 1 ;1x >2x − 1 > 0x > 1x + 1 > 0 x > −12) log x −1 (x + 1) существует при  x − 1 ≠ 1 ;  x ≠ 2 ; 1 < x < 2 и x > 2 .289. 9x + 9a(1 − a)3x − 2 − a 3 = 0; 9x + 9a(1 − a)3x − a 3 = 0; t = 3x ;t 2 + a(1 − a)t − a 3 = 0; t1,2 =a2 − a ± a2 + a2.При a>0, a=–1, то x=log3a2; если a<0, a ≠ −1, то x1=log3a2, x2=log3(–a).290.

1) log10 5 + log10 2 = log10 5 ⋅ 2 = log10 10 = 1 ;2) log10 8 + log10 125 = log10 8 ⋅125 = log10 10 3 = 3 ⋅ log10 10 = 3 ;3) log12 2 + log12 72 = log12 2 ⋅ 72 = log12 12 2 = 2 ⋅ log12 12 = 2 ;4) log 3 6 + log 333= log 3 6 ⋅ = log 3 32 = 2 log 3 3 = 2 .22291. 1) log 2 15 − log 21515= log 2 15 ⋅ = log 2 2 4 = 4 ⋅ log 2 2 = 4 ;161689www.5balls.ru2) log 5 75 − log 5 3 = log 575= log 5 5 2 = 2 ⋅ log 5 5 = 2 ;3−35411= log 1   = −3 ⋅ log 1 = −3 ;233333 3 311− log 8 32 = log 8= log 8 8 −3 = −3 ⋅ log 8 8 = −3 .4) log 81616 ⋅ 32222292. 1) log13 5 169 = log13 135 = log13 13 = ;552222) log11 3 121 = log11113 = log1111 = ;333) log 1 54 − log 1 2 = log 1−55111 43) log 1 243 = log 1   = − log 1 = −1 ;4 33433  37−171= log 2 2 6 = − log 2 2 = −1 .4) log 2 666128312 ⋅ 2041= log8 8 4 = log8 8 = 1 ;293. 1) log8 12 − log8 15 + log8 20 = log81533315 ⋅ 18312) log9 15 + log9 18 − log9 10 = log9= log9 9 2 = log9 9 = 1 ;1022131333) log 7 36 − log 7 14 − 3log 7 21 = log 7 36 2 − log 7 14 − log7 21 =26= log 7 6 − log 7 14 − log 7 21 = log 7= −2 ⋅ log 7 = −2 ;14 ⋅ 21114) 2log 1 6 − log 1 400 + 3log 1 3 45 = log 1 62 − log 1 400 2 +2333334( )+ log 13( 3 45 )3294.

1)−4136 ⋅ 451= log 1   = −4log1 = −4 .32033 3 3= log 1 36 − log 1 20 + log 1 45 = log 13333log 3 8 log 3 23 ⋅ log 3 2 3=== ;log 3 16 log 3 2 4 4 ⋅ log 3 2 42)log 5 27 log 5 33 3 log5 3 31=== =1 ;2log 5 92 log 5 3 22log 5 33)log5 36 − log5 12 log5 12log5 3 1=== ;log 5 9log5 32 2log 5 3 24)log 7 8log 7 23 3 ⋅ log 7 2 −3 ⋅ log 7 2==== −3 .log15 − log 7 30 log 15 log 7 2−11 ⋅ log 7 27363090www.5balls.ru295.

1) loga x = loga (a 3b2 c) = loga a 3 + loga b2 + loga c == 3 log a a + 2 log a b +2) loga x = loga11log a c = 3 + 2 ⋅ 3 + (−2) = 8 ;22a43 bc3= loga a 4 + loga 3 6 + loga c−3 =11= 4 log a a + log a 6 − 3 ⋅ log a c = 4 + ⋅ 3 − 3 ⋅ (−2) = 11 .33241log 2log 2 24 − log 2 72722296. 1)===3118log 3 18 − log 3 72 log 3 18 − log 3 72 log 3337233log 2 2 2 2 log 2 2 91=== =13388log3 3log3 3 44141log 7log 7 14 − log 7 563 56log 7 14 − log 7 3 563log 2 24 − log 2 722)12log 6 30 − log 6 15022=log 7 7 3log 6 612=3⋅ log 7 7=1⋅ log 6 62=log 6 30 − log 6 150=log 6=3015041=1331log 2 22 + log 2 (2 − 5)23) log 2 4 + log 2 10 ==2log 2 20 + 3log 2 2log 2 2 + 3112log 2 2 + (log 2 2 − log 2 5 )(5 + log 2 5) 12==2=2log 2 2 + log 2 5 + 35 + log 2 52;113log 7 2 − log 7 26022== 0.=115log3524log5 2 + log5 27 4log5 2 + log5 333297.

1) log3 x = 4log3 a + 7 log3 b = log3 a4 + log3 b7 = log3 a4 ⋅ b74)3log 7 2 − log 7 64х=а4b7;2) log5 x = 2 log5 a − 3 log5 b = log5 a 2 − log5 b3 = log5= log3 (a 4 ⋅ b7 );a2a2bb3; x=3;21213) log 1 x = log 1 a − log 1 b; log 1 x = log 1 a 3 − log 1 b 5 ;352222222log 1 x = log 1 (22a31);b591www.5balls.ru441141144) log 2 x = log 2 a + log 2 b = log 2 a 4 + log 2 b 7 = log 2 a 4 ⋅ b 7 ; x = a 4 ⋅ b 7 .4733333392www.5balls.ru()= 52 +102298. 1) 36log 6 5 + 101−log10 2 − 8log 2 3 = 6log 6 5 +10log10 2(− 2log 2 3)=310 3− 3 = 25 + 5 − 27 = 3 ;21 1− log922) (814×(7log7 2 )2 = (941+ 25log125 8 ) ⋅ 49log7 2 = (9 23log 49− log942+ (125log125 8 ) 3 ) ×23+ 8 3 ) ⋅ 22 = ( + 4) ⋅ 4 = 3 + 16 = 19 ;413) 161+ log4 5 + 4 2log 32+ 3log5 5 = 16 ⋅ (4log4 5 ) 2 + 2log2 3 ⋅ (8log8 5 ) 2 == 16 ⋅ 52 + 3 ⋅ 52 = 19 ⋅ 25 = 475 ;14) 72 ⋅ (49 2log7 9 − log7 6+5− log54  log7 9  1) = 72 ⋅   7+  (7log7 6 )2   log5452 = 72 ⋅  9 + 1  = 36 16 172 9= 72 ⋅  +  = 18 += 22,5 .16 36 16 11111299.

a loga pb = (a ploga pb ) p = b p = (a loga b ) p = a p loga b , значит, logap b = loga b ;p1111) log 36 2 − log 1 3 = log 62 2 − log 6−1 3 = log 6 2 2 − log 6 −1 3 =2226=11111log 6 2 − log 6 3 = log 6 (2 ⋅ 3) = log 6 6 = ;2222230= log5 5 = 1 .6300. 1) log 3 50 = log 1 (50 ) = 2 log 3 50 = 2(log 3 5 + log 3 10 ) =2) 2log2530+ log0,2 6 = 2log52 (30) + log5−1 (6) = log5 30+ log5 6 = log532= 2(log3 3 + log3 5 + log3 10 − 1) = 2(log3 15 + log3 10 − 1) = 2(a + b − 1) ;111(log 2 54 + log 2 2) = 2log 2 5 + = 2a + .2222) lg 7 ≈ 0,845 ;24) lg ≈ −0,176 .32) ln 2 ≈ 0,693 ;64) ln ≈ −0,154 .7lg 82) log5 8 =≈ 1,65 ;≈ 1,29 ;lg 52) log 4 1250 = log 22 (54 ⋅ 2) =301. 1) lg 23 ≈ 1,362 ;3) lg 0,37 ≈ −0,432 ;302. 1) ln 81 ≈ 4,394 ;3) ln 0,17 ≈ 1,772 ;303.

1) log7 25 =lg 25lg 73) log9 0,75 = lg 0,75 ≈ −0,13 ;lg 94) log0,75 1,13 = lg1,13 ≈ −0,42 .92www.5balls.rulg 0,752) log8 15 = ln 15 ≈ 1,3 ;304. 1) log7 5 = ln 5 ≈ −0,83 ;ln 7ln 93) log0,7 9 =≈ −6,16 ;ln 0,7305. 1) log5 3 = log7 3 ;log7 5ln 84) log1,1 0,23 = ln 0,23 ≈ −15,42 .ln1,1log672) lg 6 =;log 7 103) log2 7 = log7 7 =1 ;log7 24) log5 1 =5) lg7 1 = log7 7 =1 ;log7 106) log3 7 = log7 7 =log7 23log7 10lg 625306. 1) 5 lg 25 = 5lg( 25)2lg 253log713log 7 5log7 3;1 .log7 32 lg 25= 5 lg 25 = 52 = 25 ;2) log 1 (log3 4 ⋅ log 2 3) = log 1 (log3 22 ⋅ log 2 3) = −4411log 2 2 = − .22307. 1) log5 x = 2 log5 3 + 4 log 25 2; log5 x = log5 32 + 4 log52 2;log5 x = log5 32 + log52 22 = log5 9 ⋅ 4; log5 x = log5 36; x = 36 ;2) log 2 x − 2log 1 x = 9; log 2 x + log 2 x 2 = 9 log 2 2; log 2 x 3 = log 2 2 9 ;2393x =2 ; x =2 =8;3) log3 x = 9 log 27 8 − 3 log3 4; log3 x = 9 log33 8 − log3 43 ; 83 log3 x = 3 log3 8 − log3 64; log3 x = log3  ; x = 8 ; 64  1224) log9 x + log x = 3; log3 x + 2 log3 x = 3 ⋅ log3 3;32log3 x + log3 x 2 = log3 33 ; log 3 x 3 = log 3 3 3 ; x 3 = 3 3 ; x = 3 ;135) log2 x + log8 x = 8; log 2 x + log 2 x = 8 log 2 2;144log 2 x + log 2 x 3 = log 2 28 ; log 2 x 3 = log 2 28 ; x 3 = 28 ; x = 64 ;6) log 4 x − log16 x = 1 ; log 4 x − 1 log 4 x = 1 log 4 4;412212412111log 4 x − log 4 x = log 4 2 ; log 4 x = log 4 2 2 ; x 2 = 2 2 ; x = 2 .1111308.