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Calculo A
Funcao Exponencial e logarıtmica 1
1. Expresse as quantidades abaixo na forma de um unico logaritmo
a) log5 a+ log5 b− log5 c
b) log2 x+ 5 log2 (x+ 1) + 12
log2 (x− 1)
c) 13
ln x− 4 ln (2x+ 3)
d) ln x+ a ln y − b ln z
2. Resolva as seguintes equacoes
a) log2 x = 3 b) 2 = log5 (x− 1) c) 3x+2 = m (m > 0) d) ln x = 2
e) ln x = ln 2 + ln 8 f) ln (e2x−1) = 5 g)m = ln(lnx)
3. Determine o domınio das funcoes
(a) f(x) = 1
16x2−2x
(b) f(x) =√
2x − 3x
(c) f(x) = log2 x2
f(x) = 2 log2 x
(d) f(x) = logx 5
(e) f(x) = 1log(100−x)
(f) f(x) = lnx+ ln(x− 1)
(g) f(x) = lnx(x− 1)
(h) f(x) = log3+x(x2 − 1)
(i) f(x) = log3(log 12x)
(j) f(x) = log(x2 + 1)
(k) f(x) = log(3x−x2x−1
)(l) f(x) =
√log3
2x−3x−1
(m) f(x) =√x+5
log(9−5x)
(n) f(x) =√x2−4
log2(x2+2x−3)
(o) f(x) = logx+1(x2 − 3x+ 2)
1(i) Ao escrevermos log x, sem especificar a base do sistema de logaritmos que estamos empregando,
assume-se que a base e qualquer numero real positivo.
(ii) Por lnx entendemos loge x, onde e = 2.71...
1
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(p) f(x) = logx log 12(43− 2x−1)
4. Determine a imagem das funcoes
(a) f(x) = 10−x2
(b) f(x) = 11−2−x
(c) f(x) = 4x − 2x + 1
(d) f(x) = log(x2 + 10)
(e) f(x) = log2(4− x4)
(f) f(x) = log3 x+ logx 3
5. Determine x solucao de log 14(x+ 1) = log4(x− 1)
6. Seja f : A→ R definida por f(x) = | ln(x2−x+1)|. Determine A de modo a termos
f injetiva.
7. Seja f(x) = ln(x2 + x + 1), x ∈ R. Determine funcoes h, g : R → R tais que
f(x) = g(x) + h(x),∀x ∈ R, sendo h uma funcao par e g uma funcao ımpar.
8. Seja a2 + b2 = 7ab. Mostre que log a+b3
= 12(log a+ log b)
9. Mostre
a) loga b logb c = loga c
b) loga b = 1logb a
10. Mostre que log2 5 e irracional. Isto e, mostre que ele nao pode ser escrito na formapq
com p, q ∈ Z.
11. Suponha que b, c, p, q sao positivos e que b/c = p/q. Mostre que ln b−ln c = ln p−ln q.
12. Seja
f(x) =ex
e2x + 1
Mostre que f e funcao par.
13. Seja f(x) = 12(ax + a−x), (a > 0). Mostrar que
f(x+ y) + f(x− y) = 2f(x)f(y)
14. Mostre que loga nlogam n
= 1 + logam
2
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15. Sejam x, y, z tal que se tenha
x(y + z − x)
log x=y(z + x− y)
log y=z(x+ y − z)
log z
Mostre que xyyx = zyyz = xzzx
16. Simplifique a expressao
alog(log a)
log a
17. Sejam y = 101
1−log10 x , z = 101
1−log10 y . Mostre que x = 101
1−log10 z
18. Sejam a, b, c numeros reais positivos satisfazendo a2 + b2 = c2. Mostre que
logb+c a+ logc−b a = 2 logc+b a logc−b a
19. Sejam a > 0, c > 0, b =√ac, a 6= 1, c 6= 1, ac 6= 1 e N > 0. Mostre que
logaN
logcN=
logaN − logbN
logbN − logcN
20. Mostre que
loga1a2...an x =1
1loga1 x
+ 1loga2 x
+ ...+ 1logan x
21. Sejam dadas
a, a1, a2, ..., an, ... : progressao geometrica de razao q > 0
b, b1, b2, ..., bn, ... : progressao aritmetica com diferenca r > 0.
Encontre a base β de um sistema de logarıtmos onde se tem
logβ an − bn = logβ a− b, ∀n ∈ N
Respostas
1. (a) log5abc
(b) log2x(x+1)5√
x−1
(c) ln3√x
(2x+3)4
(d) ln xya
zb
2. (a) x = 8
(b) x = 26
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(c) x = −2 + log3m
(d) x = e2
(e) x = 16
(f) x = 3
(g) x = eem
3. (a) R− {0, 14}
(b) (−∞, 0]
(c) R− {0}(0,∞)
(d) (0, 1) ∪ (1,∞)
(e) (−∞, 99) ∪ (99, 100)
(f) (1,∞)
(g) (−∞, 0) ∪ (1,∞)
(h) (−3,−2) ∪ (−2,−1) ∪ (1,∞)
(i) (0, 1)
(j) R
(k) (−∞, 0) ∪ (1, 3)
(l) (−∞, 1) ∪ [2,∞)
(m) [−5, 85) ∪ (8
5, 95)
(n) (−∞,−1−√
5) ∪ (−1−√
5,−3) ∪ [2,∞)
(o) (−1, 0) ∪ (0, 1) ∪ (2,∞)
(p) (0, 1) ∪ (1, 1 + log243)
4. (a) (0, 1]
(b) (−∞, 0) ∪ (1,∞)
(c) [34,∞)
(d) [1,∞)
(e) (−∞, 2]
(f) (−∞,−2] ∪ [2,∞)
5.√
2
4
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6. A = [0, 12] ou A = (−∞, 0], ou A = [1
2, 1] ou A = [1,∞)
7. g(x) = 12
ln(x2+x+1x2−x+1
)h(x) = 1
2ln(x4 + x2 + 1)
8.
9.
10.
11.
12.
13.
14.
15.
16. log a
17.
18.
19.
20.
21. β = q1r
5
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