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Câu hỏi: Cho $a,b,c$ là các số thực khác $0$ thỏa mãn ${{4}^{a}}={{25}^{b}}={{10}^{c}}.$ Giá trị $T=\dfrac{c}{a}+\dfrac{c}{b}$ là:
A. $T=\dfrac{1}{2}$.
B. $T=\dfrac{1}{10}$.
C. $T=2$.
D. $T=\sqrt{10}$.
Ta có ${{4}^{a}}={{25}^{b}}={{10}^{c}}\Leftrightarrow \left\{ \begin{aligned}
& {{10}^{c}}={{4}^{a}} \\
& {{10}^{c}}={{25}^{b}} \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& c=\log {{4}^{a}} \\
& c=\log {{25}^{b}} \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& c=a\log 4 \\
& c=b\log 25 \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& \dfrac{c}{a}=\log 4 \\
& \dfrac{c}{b}=\log 25 \\
\end{aligned} \right.$.
Vậy $T=\dfrac{c}{a}+\dfrac{c}{b}=\log 4+\log 25=\log 100=2$.
A. $T=\dfrac{1}{2}$.
B. $T=\dfrac{1}{10}$.
C. $T=2$.
D. $T=\sqrt{10}$.
Ta có ${{4}^{a}}={{25}^{b}}={{10}^{c}}\Leftrightarrow \left\{ \begin{aligned}
& {{10}^{c}}={{4}^{a}} \\
& {{10}^{c}}={{25}^{b}} \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& c=\log {{4}^{a}} \\
& c=\log {{25}^{b}} \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& c=a\log 4 \\
& c=b\log 25 \\
\end{aligned} \right.\Leftrightarrow \left\{ \begin{aligned}
& \dfrac{c}{a}=\log 4 \\
& \dfrac{c}{b}=\log 25 \\
\end{aligned} \right.$.
Vậy $T=\dfrac{c}{a}+\dfrac{c}{b}=\log 4+\log 25=\log 100=2$.
Đáp án C.