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Getal & Ruimte vwo B Hoofdstuk 9
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Getal & Ruimte vwo B Hoofdstuk 9
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9 Exponentiële en logaritmische functiesVoorkennis Exponenten
translatie ( l , 2)
y = GY
b y
--------- ----------------------y =-3
B 1 = (-3,~) en Bg = (2,~).
c De optie zero (Tl) of ROOT (Casio) geeftx ~ 2,71.
f(x) :::: 0 geeft x:::: 2,71
d g(O) = Gt
1
+ 2 = ~ + 2 = 3~
Voor x:::: 0 is 2 < g(x):::; 3~ .
e Intersect geeft x ~ 4,07.
f(x) '.S g(x) geeft x '.S 4,07
f B 1 = (-3 , ~), dusf(x) = p heeft geen oplossingen voor p :S-3.
g AB= g(2) -/(2) = 2,6 - - 0,75 = 3,35
h Voer iny 3 = 5.
Intersect mety 1 eny2 geeftx = 5,128".
Intersect met y 1 eny3 geeftx = - 1,150".
CD= 5,128". - -1 ,150". ~ 6,28
0 a y = 4X
l trans latie ( 1, 3)
f(x) = 4 x- l +3
De horizontale asymptoot is de lijny = 3.
by = GY
t verm. x-as, 6
y = 6· GY
l trans latie (0, - 1)
g(x) = 6 · GY - 1
De horizontale asymptoot is de lijn y = - 1.
C y = 6X
t verm. x-as, }
y=~ · 6X
l trans latie (- 3, - 2)
Y = ~ . 6x+3 - 2
l verm. y -as, - 1
Y = ~ . 6-x+3 - 2
Dus g (x) = ~ · 63 - x - 2.
De horizontale asymptoot is de lijny "" -2.
4 Hoofdstuk 9 © Noordhoff Uitgevers bv
,0 a 2 + 3 · 2 2' -
1
= 98 f 2x + 1 - 2x - 1 = 96
3 · 2 2x- I = 96 2x·2 1 - 2X·2-I =96
22x- 1=32 2 · 2x - ! ·2x = 96
22x -1=2s 1!·2X=96
2x-1 = 5 2x= 64
2x = 6 2x = 26
x=3 x= 6
b 34x- 1 = _L '3
27V J g
4x2 + 2 = 3x2- 1
2 2 2 1
34x-1 = 3-J.321 (12r + = (23r -
34x- 1 = 3-2t 22x1 +4 = 23x2 - 3
4x - 1 =-2! 2x2 + 4 = 3x2 - 3
4x = - 1! - x2 = - 7
_ _ ;i_
X - 8 x2 = 7
C 4.2x- J= 8X x = ..fi v x = -..fi
22. 2x- 3 = (1 3t h 25 · y- I = 5 · 0 2x
2x- I = 23x 52.5x- 1=5 ·(kt
x- 1=3x sx + i = s. (s-1r
- 2x = 1 5x+ I = 51. 5-x
x= - l
2 x 5x+ l = 51 - x
d l. 9x = 9 x· ( 13 )
3
3-1. (32) = 32. (3-1)
x x+ l = l-x
3-I. 32x = 3 2 · 3-x 2x = O
32x- I = 32- x x=O
32 · 2x- l = l. 3x
2x - 1 =2-x 4
2s . 2x- I = 2-2. (23)
x
3x= 3 2x+ 4 = 2-2. 23x
x=l 2x+4 = 23x-2
e 2x+ 1 + 2x - 1 = 80
x + 4 = 3x - 2
2x · 2 1 + 2x · 2-I = 80
- 2x = - 6
2 · 2x + ! ·2x = 80
x=3
2! · 2x = 80
2x= 32
2x = 2s
x=5
Bladzijde 10
0 a gdag = 1,3
gweek = 1,37:.:::: 6,27
Het groeipercentage per week is 527%.
b gdag = 1,3
1
guur = 1,324:.:::: 1,011
Het groeipercentage per uur is 1, 1%.
0 a gdag = 0,95
gweek = 0,95 7 :.: : 0,698
De afname is 30,2% per week.
b gjaar = 1,0475
g lOjaar = 1,0475 10 :.: : 1,591
De toename per 10 jaar is 59,1 %.
C gweek = 2,4
1
gdag = 2,47:.: : 1,133
De toename per dag is 13,3%.
A 600 1
v N = b · g 1 met g 3 dagen = = 1,2, dus gdag = 1,23 = 1,0626".
500
l,0~2 "·
1
N_= b. } b · 1,062". 2 = 500
t - 2enN - 500
500
b= :.: : 443
1,062". 2
Dus N = 443 · 1,063 1•
© Noordhoff Uitgevers bv Exponentiële en logaritmische functies 5
, 1000 2 (2) 1
0 N = b. gt met g1ctagen = 1500 = 3' dus gctag = 3 ' = 0,943".
N_= b. 0, 9~3 ··· } b · 0,943 ... 8 = 1500
1
t- 8 enN - 1500
1500
b= ;:::; 2384
0,943 ... 8
Dus N = 2384 · 0,9441•
9.1 Logaritmen
Bladzijde 11
0 a 23 = 8 e 3-3 = ...L
27
b 2-2 = ~ f 3~ = <j3
e a 5 log(l25) = 5log(5 3 ) = 3
10
b 1og(O,l) = = -110
1og(lo- 1)
d 7 log(49) = 7log(7 2) = 2
e 2 log(~) = 21og(2i) = ~
g 4 log(0,25) = 4 log(4- 1) = - 1
h 4 log(4) = 4 log(4 1) = 1
c 2log(4) = 2log(22) = 2 f 2log(0,5) = 21og(2- 1) = - 1 i 4 log(l) = 4 log(4°) = 0
Bladzijde 12
0 a 2Jog( 64~ = 2Jog(2 6 · 2!) = 2Jog(26t) = 6~
b 3 Jog(~/3) = 3Jog(3 -2. 3!) = 3 Jog(3- 1!) = -1~
3 21 5
c log(3 · ) = 21,5
d 5 log{i ~ 5 ) = 5 log(S-3) = -3
3
e hog( i? ) = hog((t) ) = 3
2
f hogG) = hog((!) ) = 2
g 2log( 3'2 · ·ifi'J = 2log(2- 5
· 2t ) = 2log(2-4~ ) = -4~
5
h log( 1) = 5 log( 5°) = 0
i 3
log( 81 · lfi?J= 3log(3 4 · if33) = 3 log(3 4 · 3~ ) = 3 log(3 4~) = 4~
t 0 a 21og(x) = 3 c 5log(x) = 12
b 3 1og(x) = -2 1
x = 23 x = 3-2 x = 5>
x=8 x--91 x= J5
0 a 3log(x + 2) = 2 c 3
1og(2x + 1) = 4 e hog(x - 1) = 3
x + 2 = 32 2x + 1 = 34 x-1 = m3
x+2 = 9 2x + l = 81 x- l =k
x =7 2x = 80 x = Ik
b 1+hog(x)=4 x = 40 f 2log(x2 - 4) = 5
hog(x) = 3 4
d 5 + log(x) = 3 x2 - 4 = 25
x=m 3 4 1og(x) = - 2 x 2 - 4 = 32
x =k x = 4-2 x 2 = 36
x -16
- 1 x =6vx= - 6
G a 4 · 3log(x) = 2 c 3 + 2log(x) = - 1 e 31og(0,4x - 5) = 2
3 log(x) = l 2log(x) = - 4 0,4x - 5 = 32
1 2
x = 32 x= 2-4 0,4x- 5 = 9
x = j3 x = _l_
16 0,4x = 14
b 3 log(4x - 1) = - 2 d 5log(3x + 2) = 1 x= 35
4x - 1 = 3-2 3x + 2 = 5 1 f 4 + 2 · 21og(x) = 7
4x - 1 = & 3x + 2 = 5 2 · 2log(x) = 3
2
4x = l& 3x = 3 log(x~ = 1~
x= 2-
18 x =l x = 2 1>
x = 2 1 • 2!
x= 2~
6 Hoofdstuk 9 © Noordhoff Uitgevers bv
, Bladzijde 13
Ga x 0 2 3
y= 2x 1-: 1-:ti 1 1 1 1 1 2 4 8
1 1 1
x 8 4 2 2 4 8
y = 2log(x) -3 -2 -1 0 2 3
b y
/
/
/
/
/
/
/
5
/
/
/
4
l =2x /
// Y=X
/
/ /
/
v
/
/
3 /
/
y= 2~
---
/
/
/
/
2
1/ ~
/
/
I/
/
/
/
v
/
/
-
/
/
/
L--" /
-1 // 0/1
/
x
-2 2 3 4 5
/
/
-1
/
/
/
/
/
/
/
/
/
-2
I
/
/
/
/
/
c De grafiek van h(x) = 2log(x) ontstaat uit de grafiek van/(x) = 2x bij spiegelen in de lijn y = x.
Bladzijde 14
Oa x
1
9
1
3 3 9
f(x) - 2 -1 0 2
y
b
f
-
-----
2
1
1
- i---
y =1 ~
v---- 1
1
/ 1
1
0
-1
/1 2 3 4 3-{3 6 7 8 9
x
I
-2
c /(x) = 112 geeft 3 log(x)1 = 112
x = 3 12
x = 3../3
j(x) :S 1! geeft 0< x :S 3../3
d J(..J3) = 3Iog(..j3) = !
/(27) = 3 log(27) = 3
Voor ..J3 :S x :S 27 is! :S f(x) :S 3 .
© Noordhoff Uitgevers bv Exponentiële en logaritmische functies 7
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