Eksponencijalna jednacina,logaritmi

Eksponencijalna jednacina

1. 2x=8

2x = 23

X=3

2. 5x=125

5x=53

x=3

3. 3x = 181

3x = 1

34

3x = 3-4

X= - 4

4. 6x = 16x = 60

X= 0

5. (34 ¿x+3= 1

(34 ¿x+3= ( 34 )0

X+3 = 0X= - 3

6. 2-x = 3√42-x = 3√22

2-x = 223

-x =23X= - 23

7. 8x = 5√16(23)x = 5√24

23x = 245

3x = 45

x = 453

=415

8. 16x = 2 (24)x = 24x = 1

X = 14

9. 125x =15

53x = 153x = - 1

X = -1310. (35 )

x = 925

(35 )x = 3

2

52

(35 )x = (35)

2

X = 2

11. ( 436)x = 2168

(22

62)x = 6

3

23

(26)2x = (62)

3

( 26 )2x

= ( 26 )−3

2x = - 3

X = −32

12. (5√121¿x2−8

(5√121¿x2−8 = (5√112)0

X2-8 = 0 a=1, b=0, c=-8x2=8

x1,2=±√8x1,2=±2√2x1=+2√2 x2=−2√2

Domaci: 1155,1156,1157,1159

13. ax-9 = 1ax−9

ax-9 = a-(x-9)

x-9 = -(x-9) x-9 = -x+9 x+x = 9+9 2x = 18 X= 9

14. 3√ax = a3x+22

ax3 = a3x+22

x3 = 3x+22

x∙2 = 3∙(3x+2)

2x = 9x+6

2x-9x = 6

-7x = 6

-x =67 x = -67

15. 100∙102x−1 = 100034

102∙ 102x-1 = (103¿34

102+2x-1 = 1094

2+2x-1 = 94 2x = 94 - 44 2x = 54

X = 54 ∙ 12

X = 58

16. (12)x ∙ (23)

x+1 ∙ (34 )x+2 = 6

(12)x ∙ (23)

x ∙ (23)1 ∙ (34 )x ∙ (34 )2 = 6

( 12 ∙ 23 ∙

34 )x ∙ 23 ∙

916 = 6

(14 )x ∙ 38 = 6

(14 )x = 638

(14 )x = 6 ∙ 83 (14 )x = 16

4-x = 42

-x = 2 X = -2

17. 4√56 ∙ x = 3√5x+2

56− x4 = 5

x+23

6−x4 = x+23 (6-x)∙3 = 4∙(x+2)

18-3x=4x+8 -3x-4x = 8-18 -7x= - 10

X=107

18. 0,125x−0,5

√8 = 8∙(0,25)1-x

¿¿ = 8∙((0,5)2)1-x

¿¿ = 23 ∙ (12 ¿2−2x

¿¿ = 23∙2-(2-2x)

2−3x+1,5

21,5 = 2

3∙2−2+2x

1

2-3x+1,5-1,5 = 23+(-2+2x)

-3x = 3-2+2x -3x-2x = 1 -5x =1

X= -15Domaci: 1166,1167

19. 21∙3x-5x+2 = 9∙3x+2 - 5x+3

5x+3- 5x+2 = 9∙3x+2- 21∙3x

(5x)∙(53)-5x∙52 = 9∙3x∙32-21∙3x

5x∙(125-25) = 3x∙(9∙9-21) 5x∙(100) = 3x∙(60) 5x: 3x = 60:100

(53 ¿x = 35

(53 ¿x = (53 ¿−1

X= - 1

20. 3x+1+ 3x = 108 3x∙(31+1) = 108 3x∙ 4 = 108 3x = 27 3x = 33

Logaritmi

ax =b aeR/{1 } beR+

log2 256 = 8 28 = 256log 100 = log10 100 = 2 102 = 100log b = log10 b

1. Resi logaritme: log7 49 = 2 72 = 49 log2 512 = 9 29 = 512

log 13 81 = log 1

3 34 = log 1

3 ∙(13 ¿4 = - 4

log 14 116 = log 14 142 = log 14 (14 ¿2 = 2 log3 127 = log3 133 = log33-3 = -3 log5 3√5 = log5 513 = 13

2. Izracunati: a) 4 log5 25+2 log3 27 -6 log3 8 b) log3 81+5 log 13 16 – 3 log2 a) 4 log6 25 + 2 log 3 27 -6 log2 8 = 4∙2+2∙3−6 ∙3 = 8+6- 18 = 14-18 = - 4

b) log3 81+5 log 12 16 – 3 log2 132 = log3 ∙ 34 + 5 log 12 ∙ 24 – 3log2 125 = 4+5∙ log 12 ∙ (12 ¿−4- 3 log2 ∙2-5 = 4+5∙(-4)-3∙(-5)= 4-20+15 = -1 3. Odredi X iz jednacina: a) log5 x=0 50=xX=1b) log2 x = -1 2-1 = x X= 12c) log 1

3 x = -2 (13 ¿−2=x X= (31 ¿2= 9 d) logx ∙ 128 = 7 X7 =128 X7 = 27 x= 2e) X3 = 1125 X3 = ¿X= 15

f) X-2 = 116

x-2= 142 x-2 = 4-2x= 4

Domaci: 1186,1187,1188,1189,1190,1191,1192,1193,1195Svojstva Logaritama1.log b = log10 b 2.loga= 03.loga a = 14.loga br = r ∙loga b

5.log as= 1s loga b 6.loga b = 1logb ∙ a7.logb c= log aclogab8.loga= b1∙b2 = loga b1+log b29. loga b1b2 = loga b1- loga b2

Logaritamske funkcijef(x) = loga x a¿0 a≠11.Domen DeR2.Nula funkcije y=0 0=loga x x= a0=1(1,0)

3.presek sa y-osomx=0 y=loga 0 = ay = 0nema resenja...4.tablica i grafik x 1y 0 -1 1 25.znak a¿1 y¿0 xe (1,∞) y¿0 xe (0,1)0¿a<1 y¿0 xe (1,∞) y¿0 xe (0,1)6.monotonosta¿1 y↑ xe R+a¿a¿1 y↓ xeR+7.asimptotaa¿0 y→ - ∞ x→0a¿a¿a y→ ∞ x→0 8.kodomen

DeRf(x)= log 13 x a¿0 a≠11.domenDeR+2.nula funkcije

y=0 0=log x → x= 103=1 (1,0)3.presek sa y-osom x=0 y=log 0 → 1y

3 = 0nema preseka...4.tablica i grafikx 1 3 13 - 1

5y 0 -1 1 2-1=log 1

3 x 5. znak 0<a<1 y<0 xe (1,∞) y>0 xe (-∞,1) 6. 0<a<1 y↓ xeR+ 7. asimptota X=0 0<a<1 y→ ∞ , x→0 8. kodomen DeRf(x)= log2 x 1. domen DeR 2. nula funkcije

y=0 0=log2 x→ x=20=1 (1,0) 3. presek sa y-osom x=0 y=log2 0→ 2y=0 4. tablica i grafik x 1y 0 -1 1 2-1=log2 x 1=log2 x 2=log2 xX= 2-1 = 12 , x=21=2 , x= 22=45. znak a¿1 y¿0 xe (1,∞) y¿<0 xe(-∞,1)6. monotonost a¿1 y↑ xeR+7. asimptota X= 0 a¿0 y→ -∞ x→0

8. kodomenDeRy= log3 x1. domen DeR+2. nule funkcije y= 0 0=log3 x→ x=30=1tacka (1,0)3. presek sa y-osom x=0 y= log3 0 0=3ynema resenja....4. tablica i grafik x 1 1

33 9y 0 -1 1 2

x= 3-1= 13 , x=31=3 , x=32= 95. znak

a¿1 y¿0 xe(1,∞ ¿ y¿0 xe(-∞,1)6. a¿1 y↑ xeR+7. asimptota- je prava y-osa a¿0 y→ -∞ x→08. kodomen DeR

y= log 12 x1. domen DeR2. nula funkcije y= 0 0=log 12 ∙ x → x=(12)0 = 1 tacka (1,0)

3. presek sa y-osom x=0 y= log 12 ∙ 0 , nema resenja...4. tablica i grafik x 1 2 12

14y 0 -1 1 2

5. znak 0¿a<1 y¿0 xe(1,∞ ¿ y¿0 x(-∞ ,1¿6. monotonost 0¿a<1 y↓ xeR 7. asimptota 0¿a<1 y→ ∞ x→ 08. kodomen DeR

Primenom pravila logaritmovanja resiti funkciju: 1. log7 49 = 72 = 22. log2 1 = 03. log2 160 = log2 5∙32 = log2 5∙25 = log2 ∙5 + log2 ∙25 = log5log2 + 5 = 5+0,698970

0,301029 = 5+2,3219038 =7,321935

4. log x100 = log10 x – log10 100 = log x – log10 102 = log x – 25. log 23 8 = log 23 = 16. log 24 4 = log 24 22 = 14 log2 ∙22= 14 ∙ 2 = 127. log7 + 3 log 2 + log 125 – (log 14+log 5) =log7+ log 23 + log 125-(log 14+log5)= log 7∙23-125-(log14

∙5)= log 7 ∙8 ∙125log14 ∙5 = log 100 = log 102 = 28. log 12 4 13 =log 1

2 ∙(22¿13 = log 12 223 = log 12 ∙ (12 ¿

−23 = -23

Logaritmuj primenom pravila:a) * = 3∙(x+y)∙ zb) x4 ∙ y5c) 3√ x87 (6+ y )

a) log2 3 (x+y)∙z = loga 3 + loga ( x+y)+loga z= b) loga x4 ∙ y5 = loga (x4∙y)-loga 5 = (loga x4 + loga y)-loga 5 = (4loga x + loga y )- loga 5 c) loga ∙ 3√ x87 (6+ y ) = loga ( x87 ∙¿ = 13(loga( x87 ∙(6+ y)¿¿= 13(loga x87 + loga (6+y))= 13 ((loga x8 – loga 7) + loga ( 6+y))=13 ((8 loga x – loga 7)+loga (6+y))

Antilogaritmuj:a) 3 log x-2(log y+log z)=3 log x-2(log∙ y ∙ z¿=log x3-log(y∙z)2=log x3

( y ∙ z )2b) 12(log x-3∙(log y -4 log 2))= 12(log x -3∙(log y-log z4))=12 (log x-3∙¿)= 12(logx – log( y

24)3)= 12(log x

¿¿) =

log( x¿¿c) -4log y+9(log(z+1)-log 10)+34 log 5

-4 log y+9∙(log z+110 )+3

4 log 5 = log y-4+log ( z+110 )9+ log(5¿34

=log y-4 ∙ ( z+110 )9 + (5¿34 = log y-4 ∙ ( z+110 )9+ 4√53Domaci: 1205

Logaritamske jednacine1. log3 (3x-5) = 0 uslov: 3x-5>0 3x-5=30 x > 53 3x-5=1 3x=6 X=2

2. log5 (2x-x2) = 0 uslov: 2x-x2 > 0 2x-x2 =1 2x-x2 = 0-x2+2x-1=0 x∙(2−x)= 0X1,2= −b±√b2−4 ac

2a x= 0 2-x =0X1,2= −2±√4−4−(−1 ) ∙(−1)

−2 x=2X1,2= −2±√4−4

−2X1,2= 1 -1 - - -x-0 - + +x-2 - - +F - + - xe(0,2) 3. log5 (x+1)-log5(2x-3) = 1 uslov: x+12x−3>0

x+12x−3 = 1 / ∙2x−3 x+1 > 0

X+1= 2x-3 x> -1 x-2x= -4 2x-3≠0

-x= -4 x≠ 32 X= 4 pripada...

X+1 - + +2x-3 - - +F + - +xe(-∞ ,−1¿U ¿,∞ ¿

4. 1

log x8 +

1log2x 8

+ 1

log4 x8 = 2

log8 x + log8 2x + log8 4x = 2log8 x∙2x ∙4 x=2

log8 8x3 = 2 8x3=82

8x3= 64

X3= 648X3= 23

X=2 Uslov: 8x3>0 X3>0 x>0

5. log3 (x2+1)-log 13 (x2+1) = 4

log3 ( x2+1)-log3−1(x2+1)=4

log3(x2+1)- 1−1log3(x2+1)=4

log3(x2+1)+ log3 (x2+1)=4log3(x2+1) ∙(x2+4)= 4 log3 (x2+1)2 = 4(x2+1)2= 34

(x2+1)2= 81 X2+1 =9 x2+1= -9X2= 8 x2= -10X=± √ 8 x=±√−10

X= ±2√2 nema resenja....

Uslov: (x2+1)2 > 0 xeRKvadratna jednacina uvek ima 2 resenja