Question

Seja f: R∗+ →R, f(x) = logx (6x² - 5x), uma função logarítmica. Determine o(s) valor(es) corretos de x para os quais a Im(f)={3}.

Answer 1

Resposta:

$\textsf{Leia abaixo}$

Explicação passo-a-passo:

$\mathsf{f(x) = log_x(6x^2 - 5x)}$

$\mathsf{Im(f) = \{3\}}$

$\mathsf{6x^2 - 5x = x^3}$

$\mathsf{x(6x - 5) = x^3}$

$\mathsf{x^2 = 6x - 5}$

$\mathsf{x^2 - 6x + 5 = 0}$

$\mathsf{\Delta = b^2 - 4ac}$

$\mathsf{\Delta = (-6)^2 - 4.1.5}$

$\mathsf{\Delta = 36 - 20}$

$\mathsf{\Delta = 16}$

$\mathsf{x = \dfrac{-b \pm \sqrt{\Delta}}{2a} \rightarrow \dfrac{6 \pm \sqrt{16}}{2} \rightarrow \begin{cases}\mathsf{\dfrac{6 + 4}{2} = \dfrac{10}{2}= 5}\\\\\mathsf{\dfrac{6 - 4}{2} = \dfrac{2}{2} = 1}\end{cases}}$

$\boxed{\boxed{\mathsf{S = \{5\}}}}$