View
260
Download
4
Category
Preview:
Citation preview
8/2/2019 LOGARTMOS - Questes Resolvidas
1/26
Questes Resolvidas
LOGARTMOS
8/2/2019 LOGARTMOS - Questes Resolvidas
2/26
1. Calcule x = log a b nos seguintes casos:
a) b = 625 e a = 5b) b = 81 e a = 1/3
c) b = 0,0001 e a = 100d) b = 432 e a = 0,125
) log 625 = 5 = 625
log = =
> 0 1
> 0
5 = 5 = 4
) log
8 1 = 1
3
= 81
3
= 3
= 4
) log 0,0001 = 100 = 0,0001
10 = 10 10 = 10
2 = 4 = 2
) log, 4 32 = 0,125 = 4 32
125
1000
= 2. 2
1
8
18
= 2. 2 12
= 2
2 = 2 3 =
9
2
= 3
2
8/2/2019 LOGARTMOS - Questes Resolvidas
3/26
2. Calcule: a) o logaritmo de 256 na base 2 2
b) o logaritmo de 9/16 na base 2 / 3.
) log 256 =
2 2
= 256
2
. 2
= 2
2. 2
= 2
2
= 2
2 = 2
2 = 2
3
2= 8 =
16
3
) log
916
=
2
3
=9
16
2
3
=3
2
2
3
=
3
2
2
3
=3
2
2
3
=2
3
= 4
8/2/2019 LOGARTMOS - Questes Resolvidas
4/26
3. Calcule o valor da expresso: log5 1 + 4log
45 + log3 (log5 125)
log 1 = 5 = 1
5 = 1 = 5 = 0
4
=
= = log
log 5 = log = 5
log log 125 =
log log 5 =
log 3 =
= 1
: + +
0 + 5 + 1 = 6
8/2/2019 LOGARTMOS - Questes Resolvidas
5/26
4. Se log 2 = a e log 3 = b, calcule em funo de a e b o valor da expresso:
E = 10 log (log 3 ) + log (1 + log 2 / log 3 )
10: log log3 + log 1 +log2
log3
4. Se log 2 = a e log 3 = b, calcule em funo de a e b o valor da expresso:
E = 10 log (log 3 ) + log (1 + log 2 / log 3 )
log + log 1 +
= log + log
+
= log .
+
= log +
= 1 0 = 10 = + = 10
8/2/2019 LOGARTMOS - Questes Resolvidas
6/26
5. Determinar o campo de existncia da expresso log x-3 (7 x).
log = =
> 0 1
> 0
3 > 0 3 1
7 > 0
I
I . > 3
II
II. 4
III
III. > 7 < 7
3
4
7
IIII
II
3 < < 7 4 3 , 4 4 , 7
8/2/2019 LOGARTMOS - Questes Resolvidas
7/26
6. Resolva as equaes:
a) log 5 (x + 3) = log 5 (x + 3)
b) log x (14 5x) = 2
c) log 1/5 (x 4x + 4) = 0
d) log 1/3 (x + 3x 1 ) =2
e) log4 [log2 (log3 x)] = 1/2
f) ( log8 x) 4 log8 x + 4 = 0
) . . : + 3 > 0 + 3 > 0
+ 3 > 0
+ 3 > 0 > 3 > 3
log + 3 = log + 3
+ 3 = + 3
= 0. 1 = 0 = 0 = 1
0 > 3
1 > 3
= 0 , 1
8/2/2019 LOGARTMOS - Questes Resolvidas
8/26
b) log x (14 5x) = 2
. . :
> 0
1
1 4 5 > 0 5 > 14
5 < 1 4
0 < 0 2
4 + 4 = 15
4 + 4 = 1
4 + 3 = 0...
= 1 = 3
= 1 , 3
d) log 1/3 (x + 3x 1 ) =2
. . :
+ 3 1 > 0
+ 3 1 =1
3
+ 3 1 = 3
+ 3 1 = 9
+ 3 1 0 = 0..
. = 5 = 2
5 + 3. 5 1 > 0
9 > 0
= 5
2 + 3. 2 1 > 0
9 > 0
= 2
= 5 , 2
8/2/2019 LOGARTMOS - Questes Resolvidas
10/26
e) log4 [log2 (log3 x)] = 1/2
. . :
> 0
log log = 4
log log = 4
log log = 2
log = 2
log = 4
= 3
= 8 1
= 81
f) ( log8 x) 4 log8 x + 4 = 0
. . :
> 0
log =
4 + 4 = 0... = 2
log = 2
= 8
= 6 4
= 64
8/2/2019 LOGARTMOS - Questes Resolvidas
11/26
7. Sabendo que a, b e c so nmeros reais e positivos e que log10 (a. b) = 12,6 e
log10 (a . c) = 0,2 , calcule log10 b/c .
log . = 12,6 . = 10,
log . = 0,2 . = 10,
Dividindo por , temos:.
. =
10,
10,
= 10
,,
= 10,
log
= log 10
, = 12,4
8/2/2019 LOGARTMOS - Questes Resolvidas
12/26
8. Resolva as equaes:
a) log (2x + 4x 4) + colog (x + 1) = log 4
b) log x + 2 log x 10 = 3
c) log 2 (x + 1) = log 2 (x 1) + 1
d) log x 2 . log x/16 2 = log x/64 2
e) log 4 (x 3) log 16 (x 3) = 1
f) log4
(log2
x) + log2
(log4
x) = 2
) ..: 2 + 4 4 > 0 + 1 > 0 > 1
log 2 + 4 4 log + 1 = log 4
log2 + 4 4
+ 1= log 4
2 + 4 4
+ 1= 4
2 + 4 4 = 4 + 4
2 = 8 = 4 = 2
= 2
= 2
2.2 +4.2 4 > 0
1 2 > 0
= 2
8/2/2019 LOGARTMOS - Questes Resolvidas
13/26
b) log x + 2 log x 10 = 3
. . :
> 0 1
log + 2 .log 10
log = 3
log =
+ 2.1
= 3
+ 2 = 3
3 + 2 = 0...
= 1 = 2
= 1 log = 1
= 1 0 = 1 0
= 2 log
= 2
= 1 0 = 100
= 10 , 100
log = 1
8/2/2019 LOGARTMOS - Questes Resolvidas
14/26
c) log 2 (x + 1) = log 2 (x 1) + 1
. . :
+ 1 > 0 > 1
1 > 0
log + 1
log 2= log
1 + log 2
log + 1
log 2
= log[
1 . 2]
log + 112 .log 2
= log 2 2
log
= . log
log + 11
2= log 2
2
2.log + 1 = log 2 2
log + 1 = log 2 2
+ 1 = 2 2
+ 2 + 1 = 2 2
2 3 = 0... = 1 = 3
= 1
= 3 3 1 > 0 8 > 0
= 3
8/2/2019 LOGARTMOS - Questes Resolvidas
15/26
d) log x 2 . log x/16 2 = log x/64 2
. . :
> 0 1, 16 64
log 2
log .
log 2
log
16
=log 2
log
64
1
log .
1
log l o g 16 =
1
log l o g 64
log =
1
. 4=
1
6 . 4 = 6
4 + 6 = 0
5 + 6 = 0... = 2 = 3
= 2 log = 2 = 2 = 4
= 4
= 3 log = 3 = 2 = 8
= 8
= 4 , 8
8/2/2019 LOGARTMOS - Questes Resolvidas
16/26
e) log 4 (x 3) log 16 (x 3) = 1
. . :
3 > 0 > 3
log 3 log 3
log 16= 1
log 3 =
2= 1
2= 1 = 2
log 3 = 2
3 = 4
3 = 1 6 = 1 9 = 19
8/2/2019 LOGARTMOS - Questes Resolvidas
17/26
f) log 4 (log 2 x) + log 2 (log 4 x) = 2
. . :
> 0
log log
log 4+ log log = 2
log log 2
+ log log = 2
log log + 2. log log = 4
log log + log log = 4
log log . log = 4
log .log
log 4
= 2
log .log
4= 16
log
=
.
4= 16
= 64
= 64
= 2
= 2
= 4
log = 4
= 2 = 1 6
= 16
8/2/2019 LOGARTMOS - Questes Resolvidas
18/26
9. Resolva as inequaes :
a) log 5 (4x 1) < log 5 3
b) log (x + 4) log (2x 2)
c) log (x 2x) 3
d) log 2 (x 3) + log 2 (x 2) < 1
) . . :4 1 > 0 >1
4
5 > 1
4 1 < 3
4 < 4
< 1
1
4
1
III
I
II
= 1
4< < 1
8/2/2019 LOGARTMOS - Questes Resolvidas
19/26
b) log (x + 4) log (2x 2)
. . :
+ 4 > 0 > 4 < 4
2 2 > 0 2 > 2 > 1
1 2 0 < < 1
+ 4 2 2 2 2 4
3 6
3 6
2
I
II
III
1
2
4
IIIIII
= 1 < 2
8/2/2019 LOGARTMOS - Questes Resolvidas
20/26
c) log (x2x) 3
. . :
2 > 0
Para resolver uma inequao logartmica, fica mais fcil quando temos, em ambos oslados da desigualdade, logaritmos na mesma base.
Do lado esquerdo temos um logaritmo na base .
Vamos escrever o 3 do lado direito como um logaritmo na base tambm:
log
= 3 = 1 2
= 2 = 8 3 = log
8
log
2 log
8
1 2 0 < < 1
2 8 2 8 0
I
II
c) log (x 2x) 3
8/2/2019 LOGARTMOS - Questes Resolvidas
21/26
2 > 0
2 8 0
I
II
: = 2
: 0 2
0
2
: < 0 > 2
> 0
: = 2 8
: 2 4
2
4
: 2 4
0
8/2/2019 LOGARTMOS - Questes Resolvidas
22/26
: < 0 > 2 : 2 4
0
2
4
2
= 2 < 0 2 < 4
2 , 0 2 , 4
8/2/2019 LOGARTMOS - Questes Resolvidas
23/26
d) log 2 (x 3) + log 2 (x 2) < 1
. . :
3 > 0 > 3
2 > 0 > 2 > 3 I
Vamos escrever o 1 do lado direito como um logaritmo na base 2 tambm:
log = 1 = 2 = 2 = 2 1 = log 2
d) log 2 (x 3) + log 2 (x 2) < 1
log 3 + log 2 < log 2
log 3 . 2
log 5 + 6 < log 2
2 > 1
5 + 6 < 2 5 + 4 < 0
8/2/2019 LOGARTMOS - Questes Resolvidas
24/26
5 + 4 < 0
=
5 + 4
: 1 4
1
4
: 1 < < 4
< 0
: > 3
1
3
4
= 3 < < 4
3 , 4
8/2/2019 LOGARTMOS - Questes Resolvidas
25/26
10. Resolva as equaes:
a) 2 x = 7
b) 9 x 7.3 x + 10 = 0
) 2 = 7 log 7 =
=log7
log2
0,845
0,301 2,8
) 3 7 . 3 + 1 0 = 0
3 =
7 + 1 0 = 0... = 2 = 5
= 2 3 = 2
= log 2= log2log3
0,3010.477 0,63
= 5 3 = 5
= log 5=
log5
log3
0,699
0.477 1,5
= 2,8
= 0,63 ; 1,5
8/2/2019 LOGARTMOS - Questes Resolvidas
26/26
ISERJ 2012Professora Telma Castro Silva
Recommended