Chọn đáp án B.

\(a,log_a1=c\Leftrightarrow a^c=1\Leftrightarrow c=0\Rightarrow log_a1=0\\ b,log_aa=c\Leftrightarrow a^c=a\Leftrightarrow c=1\Rightarrow log_aa=1\\ c,log_aa^c=b\Leftrightarrow a^b=a^c\Leftrightarrow b=c\Rightarrow log_aa^c=c\\ d,a^{log_ab}=c\Leftrightarrow log_ab=log_ac\Leftrightarrow b=c\Rightarrow a^{log_ab}=b\)

\(a^2+4b^2=23ab\Rightarrow a^2+4ab+4b^2=27ab\Rightarrow\left(a+2b\right)^2=27ab\)

\(\Rightarrow\dfrac{\left(a+2b\right)^2}{9}=3ab\)\(\Rightarrow\left(\dfrac{a+2b}{3}\right)^2=3ab\)

Lấy logarit cơ số c hai vế:

\(log_c\left(\dfrac{a+2b}{3}\right)^2=log_c\left(3ab\right)\)

\(\Rightarrow2log_c\dfrac{a+2b}{3}=log_c3+log_ca+log_cb\)

\(\Rightarrow log_c\dfrac{a+2b}{3}=\dfrac{1}{2}\left(log_ca+log_cb+log_c3\right)\)

\(a,a^{log_ab^{\alpha}}=c\Leftrightarrow log_ac=log_ab^{\alpha}\Leftrightarrow c=b^{\alpha}\Rightarrow a^{log_ab^{\alpha}}=b^{\alpha}\\ a^{\alpha log_ab}=c\Leftrightarrow\alpha log_ab=log_ac\Leftrightarrow log_ab^{\alpha}=log_ac\Leftrightarrow b^{\alpha}=c\Rightarrow a^{\alpha log_ab}=b^{\alpha}\\ \Rightarrow a^{log_ab^{\alpha}}=a^{\alpha log_ab}\)

\(b,a^{log_ab^{\alpha}}=a^{\alpha log_ab}\\ \Rightarrow log_ab^{\alpha}=\alpha log_ab\)

a) \(\log_{12}12^3=3.\log_{12}12=3.1=3\)

b) \(\log_{0,5}0,25=\log_{2^{-1}}2^{-2}=\dfrac{-2}{-1}\log_22=2.1=2\)

c) \(\log_aa^{-3}=-3.\log_aa=-3.1=-3\)

a: \(log_{12}12^3=3\)

b: \(=log_{0.5}0.5^2=2\)

c: \(log_aa^{-3}=-3\)

\({a^{\frac{1}{2}}} = b \Leftrightarrow {\log _a}b = \frac{1}{2} \Leftrightarrow 2{\log _a}b = 1\)

Chọn B.

\(\dfrac{a^2\cdot\sqrt[3]{a}\cdot\sqrt[5]{a^4}}{\sqrt[4]{a}}=\dfrac{a^2\cdot a^{\dfrac{1}{3}}\cdot a^{\dfrac{4}{5}}}{a^{\dfrac{1}{4}}}=\dfrac{a^{\dfrac{47}{15}}}{a^{\dfrac{1}{4}}}=a^{\dfrac{173}{60}}\)

\(\Rightarrow log_a\left(\dfrac{a^2\cdot\sqrt[3]{a}\cdot\sqrt[5]{a^4}}{\sqrt[4]{a}}\right)=log_a\left(a^{\dfrac{173}{60}}\right)=\dfrac{173}{60}\)

\(a^{2log_a\left(\dfrac{\sqrt{105}}{30}\right)}=a^{log_a\left(\dfrac{7}{60}\right)}=\dfrac{7}{60}\)

Vậy \(B=\dfrac{173}{60}+\dfrac{7}{60}=\dfrac{180}{60}=3\)