4.3  Formules met logaritmen >
Regels voor het rekenen met logaritmen
1
a

Zie hieronder:

2 2 log ( 3 ) + 2 log ( 7 )

=

Regel 1 toepassen

2 2 log ( 3 ) 2 2 log ( 7 )

=

Regel 5 toepassen

3 7

=

21

b

Als 2 t = 21 , dan t = 2 log ( 21 ) .

c

24

d

7 7 log ( 3 ) + 7 log ( 8 )

=

Regel 1 toepassen

7 7 log ( 3 ) 7 7 log ( 8 )

=

Regel 5 toepassen

3 8

=

24

2

Zie hieronder:

2 2 log ( 15 ) 2 log ( 3 )

=

Regel 2 toepassen

2 2 log ( 15 ) : 2 2 log ( 3 )

=

Regel 5 toepassen

15 : 3

=

5

3

De eerste gelijkheid volgt uit regel 3 en de tweede uit regel 5.

4
a

3 , 2 en 4 .

b

De eerste gelijkheid volgt uit regel 3.
Verder geldt (regel 5): 10 log ( 2 ) = 2 , en (weer regel 5): 2 2 log ( 8 ) = 8 .

Regels toepassen
5
a

1,161

2,322

1,161

2,161

1,161

0,161

b

Van links naar rechts naar beneden.
4 log ( 25 ) = 2 4 log ( 5 ) = 2,322
4 log ( 1 5 ) = 1 4 log ( 5 ) = 1,161
4 log ( 20 ) = 4 log ( 5 ) + 4 log ( 4 ) = 2,161
1 4 log ( 5 ) = log ( 5 ) log ( 4 ) = 1,161 want log ( 5 ) log ( 4 ) = 1,161
1 4 log ( 1 1 4 ) = 1 4 log ( 1 4 ) + 1 4 log ( 5 ) = 1 - 1,161 = 0,161

6

x = log ( 11 ) log ( 7 ) 1,232

x = log ( 7 ) log ( 11 ) 0,812

x = 1 + log ( 11 ) log ( 2 ) 4,459

x = log ( 11 ) log ( 2 ) 1,860 of x = 1,860

x = ( log ( 111 ) log ( 2 ) ) 2 46,164

x = 2 log ( 111 ) log ( 2 ) 13,589

x = 0,000

x = log ( 11 ) log ( 1,1 ) 25,159

7
a

K + 0,1 K = 1,1 K , dus de groeifactor per jaar is 1,1 .

b

K ( t ) = 5432 1,1 t

c

1,1 t = 10.000 5432 , dus t = 1,1 log ( 10000 5432 ) = 6,403069...  jaar, dus na 77  maanden.

8
a

20 % minder licht per cm, dus 80 % blijft over.
De vermenigvuldigingsfactor per cm = 0,8 en 0,8 0,8 = 0,64 .

b

100 0,8 x %

c

0,8 x = 0,4 , dus x = log ( 0,4 ) log ( 0,8 ) = 4,1  cm, dus 41  mm.

9
a

6 1,019 12 = 7,52 miljard

b

1,019 x = 2 x = log ( 2 ) log ( 1,019 ) 37 jaar

10
a

0,712

0,712

1,222

1,222

b

( a x ) 1 = a x
Dus x en y zijn elkaars tegengestelde.

11
a

6,66

2,22

0,36

0,12

b

3 y = x
8 y = ( 2 3 ) y = 2 3 y = 2 x x = 3 y

12
a
  1. 5 log ( 625 ) + 5 log ( 1 5 ) = 4 + 1 = 3

  2. 5 log ( 625 ) + 5 log ( 1 5 ) = 5 log ( 125 ) = 3

b

log ( 20 ) + log ( 5 ) = 1,3 + 0,7 = 2
log ( 20 ) + log ( 5 ) = log ( 100 ) = 2 en
log ( 5 ) log ( 1 2 ) = 0,7 0,3 = 1
log ( 5 ) log ( 1 2 ) = log ( 10 ) = 1

13

= 3 log ( 9 ) = 2

= 5 log ( 1 5 ) = 1

= 1 4 log ( 4 ) = 1

= 30 log ( 30 ) = 1

= 2 log ( 1 4 ) = 2

= 2 log ( 1 ) = 0

14

= 4 log ( 64 ) = 3

= 5 log ( 5 ) = 1

= 0,7 log ( 1 0,7 ) = 1

= 7 log ( 84 x 12 x ) = 7 log ( 7 ) = 1

= 5 log ( 6 5 3 2 ) = 5 log ( 1 5 ) = 1

15
a

In beide gevallen vind je 1,5 .

b

4 log ( 2 11 ) = 11 4 log ( 2 ) = 11 1 2 = 5 1 2

3 log ( ( 1 9 ) 11 ) = 11 3 log ( 1 9 ) = 11 2 = 22

1 4 log ( 2 11 ) = 11 1 4 log ( 2 ) = 11 1 2 = 5 1 2

1 3 log ( ( 1 9 ) 11 ) = 11 1 3 log ( 1 9 ) = 11 2 = 22

c

g log ( 1 x ) = g log ( x 1 ) = 1 g log ( x )

16
  1. a log ( b 2 ) = 2 a log ( b ) = 2 5 = 10

  2. a log ( b c ) = a log ( b ) + a log ( c ) = 5 + 3 = 8

  3. a log ( b c ) = a log ( b ) + a log ( c ) = 5 + 1 2 a log ( c ) = 5 + 1 1 2 = 6 1 2

  4. a log ( b 2 c 3 ) = 2 a log ( b ) + 3 a log ( c ) = 2 5 + 3 3 = 19

  5. a log ( b c ) = a log ( b ) a log ( c ) = 5 3 = 2

  6. a log ( 1 c ) = a log ( 1 ) a log ( c ) = 0 3 = 3

  7. a log ( 1 b 3 ) = a log ( 1 ) a log ( b 3 ) = 0 3 5 = 15

  8. a log ( b c ) = 1 2 ( a log ( b ) + a log ( c ) ) = 1 2 ( 5 + 3 ) = 4

17
a

log ( a b ) + log ( b c ) + log ( c a ) = log ( a b b c c a ) = log ( ( a b c ) 2 ) = 2 log ( a b c )

b

log ( a b ) + log ( b c ) + log ( c a ) = log ( a b c a b c ) = log ( 1 ) = 0