Question

resolva: A) log25 125 = B) log3 81 = C) log4 √2 = D) log3 1/81 = (1/8 significa divisão) E) log16 ∛8 = F) log7 343 = G) log5 625 = OBS: PRESCISO DOS CALCULOS

Answer 1

Nestes exercícios, vamos utilizar a fatoração dos logaritmandos e bases, aplicar a definição de logaritmo e propriedades de potência.

a)

$\log_{_{25}}125~=~x\\\\\\\underline{Fatorando}~o~logaritmando~e~a~base\\\\\\log_{_{5^2}}5^3~=~x\\\\\\Aplicando~a~\underline{de finicao~de~logaritmo}\\\\\\5^3~=~\left(5^2\right)^x\\\\\\Aplicando~a~propriedade~da~\underline{potencia~de~potencia}\\\\\\5^3~=~5^{2x}\\\\\\Igualdade~de~potencias~de~mesma~base,~igualamos~os~expoentes\\\\\\5\!\!\!\backslash^3~=~5\!\!\!\backslash^{2x}\\\\\\3~=~2x\\\\\\\boxed{x~=~\dfrac{3}{2}~~ou~~1,5}$

b)

$\log_{_{3}}81~=~x\\\\\\\underline{Fatorando}~o~logaritmando\\\\\\log_{_{3}}3^4~=~x\\\\\\Aplicando~a~\underline{de finicao~de~logaritmo}\\\\\\3^4~=~3^x\\\\\\Igualdade~de~potencias~de~mesma~base,~igualamos~os~expoentes\\\\\\3\!\!\!\backslash^4~=~3\!\!\!\backslash^{x}\\\\\\\boxed{x~=~4}$

Nos próximos, vou omitir os textos na resolução, mas, caso fique alguma duvida, deixe um comentário.

c)

$\log_{_{4}}\sqrt{2}~=~x\\\\\\log_{_{2^2}}2^{\frac{1}{2}}~=~x\\\\\\2^{\frac{1}{2}}~=~\left(2^2\right)^x\\\\\\2^{\frac{1}{2}}~=~2^{2x}\\\\\\2\!\!\!\backslash^{\frac{1}{2}}~=~2\!\!\!\backslash^{2x}\\\\\\\frac{1}{2}~=~2x\\\\\\\boxed{x~=~\dfrac{1}{4}~~ou~~0,25}$

d)

$\log_{_{3}}\dfrac{1}{81}~=~x\\\\\\\log_{_3}\dfrac{1}{3^4}~=~x\\\\\\log_{_{3}}3^{-4}~=~x\\\\\\3^{-4}~=~3^x\\\\\\3\!\!\!\backslash^{-4}~=~3\!\!\!\backslash^{x}\\\\\\\boxed{x~=\,-4}$

e)

$\log_{_{16}}\sqrt[3]{8}~=~x\\\\\\log_{_{2^4}}2~=~x\\\\\\2^{1}~=~\left(2^4\right)^x\\\\\\2^1~=~2^{4x}\\\\\\2\!\!\!\backslash^1~=~2\!\!\!\backslash^{4x}\\\\\\1~=~4x\\\\\\\boxed{x~=~\dfrac{1}{4}~~ou~~0,25}$

f)

$\log_{_{7}}343~=~x\\\\\\log_{_{7}}7^3~=~x\\\\\\7^3~=~7^x\\\\\\7\!\!\!\backslash^{3}~=~7\!\!\!\backslash^{x}\\\\\\\boxed{x~=~3}$

g)

$\log_{_{5}}625~=~x\\\\\\log_{_{5}}5^4~=~x\\\\\\5^4~=~5^x\\\\\\5\!\!\!\backslash^4~=~5\!\!\!\backslash^{x}\\\\\\\boxed{x~=~4}$