Question

(PUC-MG) se log a b = -2 e AB = 3,1 então b-a é igual a:
a)20/3
b)22/3
c)24/3
d)26/3​

Answer 1

Aplicando a definição de logaritmo em  $log_{_a}b~=\,-2$, teremos:

$b~=~a^{-2}\\\\\\\boxed{b~=~\dfrac{1}{a^2}}\\\\\\Vamos~agora~manipular~essa~equacao~de~modo~a~evidenciar\\o~termo~ab,~que~temos~o~valor\\\\\\b~=~\dfrac{1}{a\cdot a}\\\\\\a\cdot b~=~\dfrac{1}{a}\\\\\\3~=~\dfrac{1}{a}\\\\\\\boxed{a~=~\dfrac{1}{3}}\\\\\\Com~o~valor~de~''a''~podemos~calcular~''b'':\\\\\\b~=~\dfrac{1}{a^2}\\\\\\b~=~\dfrac{1}{\left(\frac{1}{3}\right)^2}\\\\\\b~=~\dfrac{1}{\frac{1}{9}}\\\\\\\boxed{b~=~9}$

Por fim, basta calcularmos a expressão solicitada:

$b-a~=~9~-~\dfrac{1}{3}\\\\\\b-a~=~\dfrac{3\cdot9~-~1}{3}\\\\\\b-a~=~\dfrac{27~-~1}{3}\\\\\\\boxed{b-a~=~\dfrac{26}{3}}~~~\rightarrow~~Letra~D$