Logaritmos (Mudança de base)

por Cesarr Qui 27 Fev 2014, 12:06

Olá.

$Logaritmos (Mudança de base) Gif.latex?Se%5C%20log_%7Bab%7Da%3D4%2C%5C%20calcule%5C%20log_%7Bab%7D%5Cfrac%7B%5Csqrt%5B3%5D%7Ba%7D%7D%7B%5Csqrt%7Bb%7D%7D%5C%20$

Eu fiz assim:

$Logaritmos (Mudança de base) Gif.latex?%5C%5Clog_%7Bab%7D%5C%20%5Csqrt%5B3%5D%7Ba%7D-log_%7Bab%7D%5C%20%5Csqrt%7Bb%7D%5C%5C%20%5C%5Clog_%7Bab%7D%5C%20a%5E%7B%5Cfrac%7B1%7D%7B3%7D%7D-log_%7Bab%7D%5C%20b%5E%7B%5Cfrac%7B1%7D%7B2%7D%7D%5C%5C%20%5C%5C%5Cfrac%7B1%7D%7B3%7D%5C%20.%5C%20log_%7Bab%7D%5C%20a-%5Cfrac%7B1%7D%7B2%7D%5C%20$

$Logaritmos (Mudança de base) Gif.latex?%5C%5C%5Cfrac%7B4%7D%7B3%7D-%5Cfrac%7B1%7D%7B2%7D%5C%20.%5C%20log_%7Bab%7D%5C%20b%5C%5C%20%5C%5C%5Cfrac%7B4%7D%7B3%7D-%5Cfrac%7B1%7D%7B2%7D%5C%20.%5C%20%5Cfrac%7Blog_%7Bb%7D%5C%20b%7D%7Blog_%7Bb%7D%5C%20ab%7D%5C%5C%20%5C%5C%5Cfrac%7B4%7D%7B3%7D-%5Cfrac%7B1%7D%7B2%7D%5C%20.%5C%20%5Cfrac%7B1%7D%7Blog_%7Bb%7D%5C%20a+log_%7Bb%7D%5C%20b%7D%5C%5C%20%5C%5C%5Cfrac%7B4%7D%7B3%7D-%5Cfrac%7B1%7D%7B2%7D%5C%20.%5C%20%5Cfrac%7B1%7D%7Blog_%7Bb%7D%5C%20a+1%7D%5C%5C%20%5C%5C%5Cfrac%7B4%7D%7B3%7D-%5Cfrac%7B1%7D%7B2$

A partir daí, eu fico fazendo mudança de base mas chego a lugar nenhum...

Agradeço a quem puder ajudar.

Spoiler:

por **Elcioschin** Qui 27 Fev 2014, 14:03

log[a.b](a) = 4 ---> (a.b)^4 = a ---> a³.b^4 = 1 ---> b = 1/a^(3/4) --->

√b = 1/a^(3/8 ) ---> I

a.b = a.[1/a^(3/4)] ---> a.b = a^(1/4) ---> II

log[a.b](∛a/√b) = log[a^(1/4)]{a^(1/3)/b^1/2} = log[a^(1/4)] a^(1/3).a^(3/8 )

= log[a^(1/4)](a^17/24) = x

a^(17/24) = [a^(1/4)]^x --> a^(17/24) = a^(x/4) --> x/4 = 17/24 --> x = 17/6