Đáp án:

\(\begin{array}{l}
a,\,\,\,\,x = {a^4}{b^7}\\
b,\,\,\,\,x = \sqrt[4]{a}.\sqrt[7]{{{b^4}}}\\
c,\,\,\,\,x = \dfrac{{{a^2}}}{{{b^3}}}\\
d,\,\,\,\,x = \dfrac{{\sqrt[3]{{{a^2}}}}}{{\sqrt[5]{b}}}\\
e,\,\,\,\,x = 3\\
f,\,\,\,\,x = \dfrac{{243}}{{50}}
\end{array}\)

Giải thích các bước giải:

ĐKXĐ:  \(x > 0,x \ne 1\)

 Ta có:

\(\begin{array}{l}
a,\\
{\log _3}x = 4{\log _3}a + 7{\log _3}b\\
 \Leftrightarrow {\log _3}x = {\log _3}{a^4} + {\log _3}{b^7}\\
 \Leftrightarrow {\log _3}x = {\log _3}\left( {{a^4}{b^7}} \right)\\
 \Leftrightarrow x = {a^4}{b^7}\\
b,\\
{\log _{\dfrac{2}{3}}}x = \dfrac{1}{4}{\log _{\dfrac{2}{3}}}a + \dfrac{4}{7}{\log _{\dfrac{2}{3}}}b\\
 \Leftrightarrow {\log _{\dfrac{2}{3}}}x = {\log _{\dfrac{2}{3}}}{a^{\dfrac{1}{4}}} + {\log _{\dfrac{2}{3}}}{b^{\dfrac{4}{7}}}\\
 \Leftrightarrow {\log _{\dfrac{2}{3}}}x = {\log _{\dfrac{2}{3}}}\left( {{a^{\dfrac{1}{4}}}.{b^{\dfrac{4}{7}}}} \right)\\
 \Leftrightarrow x = {a^{\dfrac{1}{4}}}.{b^{\dfrac{4}{7}}}\\
 \Leftrightarrow x = \sqrt[4]{a}.\sqrt[7]{{{b^4}}}\\
c,\\
{\log _5}x = 2{\log _5}a - 3{\log _5}b\\
 \Leftrightarrow {\log _5}x = {\log _5}{a^2} - {\log _5}{b^3}\\
 \Leftrightarrow {\log _5}x = {\log _5}\dfrac{{{a^2}}}{{{b^3}}}\\
 \Leftrightarrow x = \dfrac{{{a^2}}}{{{b^3}}}\\
d,\\
{\log _{\dfrac{1}{2}}}x = \dfrac{2}{3}{\log _{\dfrac{1}{2}}}a - \dfrac{1}{5}{\log _{\dfrac{1}{2}}}b\\
 \Leftrightarrow {\log _{\dfrac{1}{2}}}x = {\log _{\dfrac{1}{2}}}{a^{\dfrac{2}{3}}} - {\log _{\dfrac{1}{2}}}{b^{\dfrac{1}{5}}}\\
 \Leftrightarrow {\log _{\dfrac{1}{2}}}x = {\log _{\dfrac{1}{2}}}\dfrac{{{a^{\dfrac{2}{3}}}}}{{{b^{\dfrac{1}{5}}}}}\\
 \Leftrightarrow x = \dfrac{{{a^{\dfrac{2}{3}}}}}{{{b^{\dfrac{1}{5}}}}}\\
 \Leftrightarrow x = \dfrac{{\sqrt[3]{{{a^2}}}}}{{\sqrt[5]{b}}}\\
e,\\
{\log _3}x + {\log _9}x = \dfrac{3}{2}\\
 \Leftrightarrow {\log _3}x + {\log _{{3^2}}}x = \dfrac{3}{2}\\
 \Leftrightarrow {\log _3}x + \dfrac{1}{2}{\log _3}x = \dfrac{3}{2}\\
 \Leftrightarrow {\log _3}x + {\log _3}{x^{\dfrac{1}{2}}} = \dfrac{3}{2}\\
 \Leftrightarrow {\log _3}\left( {x.{x^{\dfrac{1}{2}}}} \right) = \dfrac{3}{2}\\
 \Leftrightarrow {\log _3}{x^{\dfrac{3}{2}}} = \dfrac{3}{2}\\
 \Leftrightarrow {x^{\dfrac{3}{2}}} = {3^{\dfrac{3}{2}}}\\
 \Leftrightarrow x = 3\\
f,\\
{\log _4}x = \dfrac{1}{3}{\log _4}216 - 2{\log _4}10 + 4{\log _4}3\\
 \Leftrightarrow {\log _4}x = \dfrac{1}{3}{\log _4}{6^3} - {\log _4}{10^2} + {\log _4}{3^4}\\
 \Leftrightarrow {\log _4}x = \dfrac{1}{3}.3{\log _4}6 - {\log _4}100 + {\log _4}81\\
 \Leftrightarrow {\log _4}x = {\log _4}6 - {\log _4}100 + {\log _4}81\\
 \Leftrightarrow {\log _4}x = {\log _4}\dfrac{{6.81}}{{100}}\\
 \Leftrightarrow {\log _4}x = {\log _4}\dfrac{{243}}{{50}}\\
 \Leftrightarrow x = \dfrac{{243}}{{50}}
\end{array}\)