Considerando log 2 = 0,3; log 5 = 0,7 e log 7 = 0,8 , qual o valor do log 700. *
5 pontos
0,035
2,8
2
2,4
Me ajudem, por favor
Answers
Answer:
Vamos utilizar as propriedades de logaritmo e a fatoração para reescrever os logaritmos de modo que possamos utilizar as informações dadas no enunciado para calcular o valor dos logaritmos.
a)
\begin{gathered}Utilizando~a~propriedade~da~\underline{troca~de~base}\\\\\\\log_{_3}2~=~\dfrac{\log2}{\log3}\\\\\\\log_{_3}2~=~\dfrac{0,30}{0,48}\\\\\\\log_{_3}2~=~\dfrac{30}{48}\\\\\\\boxed{\log_{_3}2~=~\dfrac{5}{8}~~ou~~0,625}\end{gathered}
Utilizando a propriedade da
troca de base
log
3
2 =
log3
log2
log
3
2 =
0,48
0,30
log
3
2 =
48
30
log
3
2 =
8
5
ou 0,625
b)
\begin{gathered}Utilizando~a~propriedade~da~\underline{troca~de~base}\\\\\\\log_{_2}5~=~\dfrac{\log5}{\log2}\\\\\\\log_{_2}5~=~\dfrac{0,70}{0,30}\\\\\\\log_{_2}5~=~\dfrac{70}{30}\\\\\\\boxed{\log_{_2}5~=~\dfrac{7}{3}~~ou~~2,333...}\end{gathered}
Utilizando a propriedade da
troca de base
log
2
5 =
log2
log5
log
2
5 =
0,30
0,70
log
2
5 =
30
70
log
2
5 =
3
7
ou 2,333...
c)
\begin{gathered}Utilizando~a~propriedade~da~\underline{troca~de~base}\\\\\\\log_{_5}3~=~\dfrac{\log3}{\log5}\\\\\\\log_{_5}3~=~\dfrac{0,48}{0,70}\\\\\\\log_{_5}3~=~\dfrac{48}{70}\\\\\\\boxed{\log_{_5}3~=~\dfrac{24}{35}}\end{gathered}
Utilizando a propriedade da
troca de base
log
5
3 =
log5
log3
log
5
3 =
0,70
0,48
log
5
3 =
70
48
log
5
3 =
35
24
d)
\begin{gathered}Utilizando~a~propriedade~da~\underline{troca~de~base}\\\\\\\log_{_3}100~=~\dfrac{\log100}{\log3}\\\\\\Fatorando~o~~ logaritmando~~''100''\\\\\\\log_{_3}100~=~\dfrac{\log\,(2\cdot2\cdot5\cdot5)}{\log3}\\\\\\Aplicando~a~propriedade~do~\underline{logaritmo~do~produto}\\\\\\\log_{_3}100~=~\dfrac{\log2~+~\log2~+~\log5~+~\log5}{\log3}\\\\\\\log_{_3}100~=~\dfrac{0,30~+~0,30~+~0,70~+~0,70}{0,48}\\\\\\\log_{_3}100~=~\dfrac{2,00}{0,48}\end{gathered}
Utilizando a propriedade da
troca de base
log
3
100 =
log3
log100
Fatorando o logaritmando
′′
100
′′
log
3
100 =
log3
log(2⋅2⋅5⋅5)
Aplicando a propriedade do
logaritmo do produto
log
3
100 =
log3
log2 + log2 + log5 + log5
log
3
100 =
0,48
0,30 + 0,30 + 0,70 + 0,70
log
3
100 =
0,48
2,00
\begin{gathered}\log_{_3}100~=~\dfrac{200}{48}\\\\\\\boxed{\log_{_3}100~=~\dfrac{25}{6}~~ou~~4,166...}\end{gathered}
log
3
100 =
48
200
log
3
100 =
6
25
ou 4,166...
e)
\begin{gathered}Utilizando~a~propriedade~da~\underline{troca~de~base}\\\\\\\log_{_4}18~=~\dfrac{\log18}{\log4}\\\\\\Fatorando~os~~ logaritmandos~~''18''~e~''4''\\\\\\\log_{_4}18~=~\dfrac{\log\,(2\cdot3\cdot3)}{\log\,(2\cdot2)}\\\\\\Aplicando~a~propriedade~do~\underline{logaritmo~do~produto}\\\\\\\log_{_4}18~=~\dfrac{\log2~+~\log3~+~\log3}{\log2~+~\log2}\\\\\\\log_{_4}18~=~\dfrac{0,30~+~0,48~+~0,48}{0,30~+~0,30}\\\\\\\log_{_4}18~=~\dfrac{1,26}{0,60}\end{gathered}
Utilizando a propriedade da
troca de base
log
4
18 =
log4
log18
Fatorando os logaritmandos
′′
18
′′
e
′′
4
′′
log
4
18 =
log(2⋅2)
log(2⋅3⋅3)
Aplicando a propriedade do
logaritmo do produto
log
4
18 =
log2 + log2
log2 + log3 + log3
log
4
18 =
0,30 + 0,30
0,30 + 0,48 + 0,48
log
4
18 =
0,60
1,26
\begin{gathered}\log_{_4}18~=~\dfrac{126}{60}\\\\\\\boxed{\log_{_4}18~=~\dfrac{21}{10}~~ou~~2,1}\end{gathered}
log
4
18 =
60
126
log
4
18 =
10
21
ou 2,1
f)
\begin{gathered}Utilizando~a~propriedade~da~\underline{troca~de~base}\\\\\\\log_{_{36}}0,5~=~\dfrac{\log0,5}{\log36}\\\\\\\log_{_{36}}0,5~=~\dfrac{\log\frac{1}{2}}{\log36}\\\\\\\log_{_{36}}0,5~=~\dfrac{\log2^{-1}}{\log36}\\\\\\Fatorando~o~~ logaritmando~~''36''\\\\\\\log_{_{36}}0,5~=~\dfrac{\log\,2^{-1}}{\log\,(2\cdot2\cdot3\cdot3)}\\\\\\Aplicando~a~propriedade~do~\underline{logaritmo~da~potencia}\end{gathered}
Utilizando a propriedade da
troca de base
log
36
0,5 =
log36
log0,5
log
36
0,5 =
log36
log
2
1
log
36
0,5 =
log36
log2
−1
Fatorando o logaritmando
′′
36
′′
log
36
0,5 =
log(2⋅2⋅3⋅3)
log2
−1
Aplicando a propriedade do
logaritmo da potencia
\begin{gathered}\log_{_{36}}0,5~=~\dfrac{-1\cdot\log\,2}{\log\,(2\cdot2\cdot3\cdot3)}\\\\\\Aplicando~a~propriedade~do~\underline{logaritmo~do~produto}\\\\\\\log_{_{36}}0,5~=~\dfrac{-1\cdot\log2}{\log2~+~\log2~+~\log3~+~\log3}\\\\\\\log_{_{36}}0,5~=~\dfrac{-1\cdot0,30}{0,30~+~0,30~+~0,48~+~0,48}\\\\\\\log_{_{36}}0,5~=~\dfrac{-0,30}{1,56}\\\\\\\log_{_{36}}0,5~=\,-\dfrac{30}{156}\\\\\\\boxed{\log_{_{36}}0,5~=\,-\dfrac{5}{26}}\end{gathered}
log
36
0,5 =
log(2⋅2⋅3⋅3)
−1⋅log2
Aplicando a propriedade do
logaritmo do produto
log
36
0,5 =
log2 + log2 + log3 + log3
−1⋅log2
log
36
0,5 =
0,30 + 0,30 + 0,48 + 0,48
−1⋅0,30
log
36
0,5 =
1,56
−0,30
log
36
0,5 =−
156
30
log
36
0,5 =−
26
5