Énoncé

1. Exprimer en fonction de log(3) :

a) A = log(81)

b) B = log

Å

1

27

ã

c) C = 5 log(27)+3 log(9)

d) D = log

Ç√3

9

å

e) E = log(√

27)

f) F = log

Å

1

7

ã

+log

Å

7

9

ã

2. Exprimer en fonction de log(2) :

a) G = log(36) − log(9)

b) H = log

Å

1

4

ã

c) I = 4 log(32)−5 log(8)

d) J = log(√

2) + log(16)

e) K = log(3) + log(4) − log(6)

3. Calculer :

a) L =log(64)

log(8)

b) M = log(103)+10log(4)

c) N = log

Å

1

8

ã

+3 log(2)

d) P = log(0,01)

e) Q = log(10000)

f) R =log(54)

log(52)

g) S = log(14) − log(28) + log(2)

h) T = 2 log(3) − 3 log(2) + log

Å

8

9

ã

Corrigé

1. a) A = log(81)= log(34)

= 4 log(3)

b) B = log

Å

1

27

ã

= − log(27)= − log(33)

= −3 log 3

c) C = 5 log(27) + 3 log(9)= 5 log(33) + 3 log(32)= 5 × 3 log(3) + 3 × 2 log(3)= 15 log(3) + 6 log(3)= (15 + 6) log(3)

= 21 log(3)

d) D = log

Ç√3

9

å

= log(√

3) − log(9)

=1

2log(3) − log(32)

=1

2log(3) − 2 log(3)

=

Å

1

2− 2

ã

log(3)

=

Å

1

2−

4

2

ã

log(3)

= −3

2log(3)

e) E = log(√

27)

=1

2log(27)

=1

2log(33)

=1

2× 3 log(3)

=3

2log(3)

f) F = log

Å

1

7

ã

+log

Å

7

9

ã

= log

Å

1

7×

7

9

ã

= log

Å

1

9

ã

= − log(9)= − log(32)

= −2 log(3)

Autre méthode :

F = log

Å

1

7

ã

+log

Å

7

9

ã

= − log(7) + log(7) − log(9)= − log(9)= − log(32)

= −2 log(3)

2. a) G = log(36) − log(9)

= log

Å

36

9

ã

= log(4)= log(22)

= 2 log(2)

b) H = log

Å

1

4

ã

= − log(4)= − log(22)

= −2 log(2)

c) I = 4 log(32) − 5 log(8)= 4 log(25) − 5 log(23)= 4 × 5 log(2) − 5 × 3 log(2)= 20 log(2) − 15 log(2)= (20 − 15) log(2)

= 5 log(2)

d) J = log(√

2) + log(16)

=1

2log(2) + log(24)

=1

2log(2) + 4 log(2)

=

Å

1

2+ 4

ã

log(2)

=

Å

1

2+

8

2

ã

log(2)

=9

2log(2)

e) K = log(3) + log(4) − log(6)= log(3 × 4) − log(6)= log(12) − log(6)

= log

Å

12

6

ã

= log(2)

3. a) L =log(64)

log(8)

=log(82)

log(8)

=2 log(8)

log(8)= 2

b) M = log(103) + 10log(4)

= 3 + 4

= 7

c) N = log

Å

1

8

ã

+3 log(2)

= − log(8) + 3 log(2)= − log(23) + 3 log(2)= −3 log(2) + 3 log(2)

= 0

d) P = log(0,01)= log(10−2)

= −2

e) Q = log(10000)= log(104)

= 4

f) R =log(54)

log(52)

=4 log(5)

2 log(5)= 2

g) S = log(14) − log(28) + log(2)

= log

Å

14

28

ã

+ log(2)

= log

Å

1

2

ã

+ log(2)

= − log(2) + log(2)

= 0

h) T = 2 log(3) − 3 log(2) + log

Å

8

9

ã

= log(32)−log(23)+log(8)−log(9)= log(9)−log(8)+log(8)−log(9)

= 0