Cho $\Large \log_{b}a=x; \log_{b}c=y$. Hãy biểu diễn $\Large \log_{a^2

Cho $\Large \log_{b}a=x; \log_{b}c=y$. Hãy biểu diễn $\Large \log_{a^2}\left(\sqrt[3]{b^5c^4}\right)$ theo x và y

• A.

  A. $\Large \dfrac{5+4y}{6x}$

   
• B.

  B.  $\Large \dfrac{20y}{3x}$

• C.

  C.  $\Large \dfrac{5+3y^4}{3x^2}$

• D.

  D.  $\Large 20x+\dfrac{20y}{3}$

Chọn A

$\Large \log_{b}a=\dfrac{\ln a}{\ln b}=x$ $\Large \Rightarrow \ln a=x\ln b (a, b>0)$

${\log _b}c = \frac{{\ln c}}{{\ln b}} = 7 \Rightarrow \ln c = y\ln b\,\left( {b,c > 0} \right)$

$ \Large \log_{a^2}\left(\sqrt[3]{b^5c^4}\right)$ $\Large =\dfrac{\ln\left(\sqrt[3]{b^5c^4}\right)}{\ln(a^2)}$ $\Large =\dfrac{\ln\left(b^{\dfrac{5}{3}}c^{\dfrac{4}{3}}\right)}{2\ln a}$ $\Large =\dfrac{\dfrac{5}{3}\ln b+\dfrac{4}{3}\ln c}{2\ln a}$ $\Large =\dfrac{\dfrac{5}{3}\ln b+\dfrac{4}{3}y\ln b}{2x\ln b}$ $\Large =\dfrac{5+4y}{6x}$