Bài 94
\(\eqalign{
& a)\,{\log _3}\left( {\log _{0,5}^2x - 3{{\log }_{0,5}}x + 5} \right) = 2\,; \cr
& c)\,1 - {1 \over 2}\log \left( {2x - 1} \right) = {1 \over 2}\log \left( {x - 9} \right)\,; \cr} \)
\(\eqalign{
& b)\,{\log _2}\left( {{{4.3}^x} - 6} \right) - {\log _2}\left( {{9^x} - 6} \right) = 1\,; \cr
& d)\,{1 \over 6}{\log _2}\left( {x - 2} \right) - {1 \over 3} = {\log _{{1 \over 8}}}\sqrt {3x - 5} . \cr} \)
\(\eqalign{
& a)\,\,{\log _3}\left( {\log _{0,5}^2x - 3{{\log }_{0,5}}x + 5} \right) = 2 \Leftrightarrow \log _{0,5}^2x - 3{\log _{0,5}}x + 5 = 9 \cr
& \Leftrightarrow \log _{0,5}^2x - 3{\log _{0,5}x} - 4 = 0 \Leftrightarrow \left[ \matrix{
{\log _{0,5}x} = - 1 \hfill \cr
{\log _{0,5}x} = 4 \hfill \cr} \right. \cr
& \Leftrightarrow \left[ \matrix{
x = {\left( {0,5} \right)^{ - 1}} = 2 \hfill \cr
x = {\left( {0,5} \right)^4} = {1 \over {16}} \hfill \cr} \right. \cr} \)
Vậy \(S = \left\{ {2;{1 \over {16}}} \right\}\)
b) Ta có: \({\log _2}\left( {{{4.3}^x} - 6} \right) - {\log _2}\left( {{9^x} - 6} \right) = 1 \Leftrightarrow {\log _2}\left( {{{4.3}^x} - 6} \right) = {\log _2}2\left( {{9^x} - 6} \right)\)
\( \Leftrightarrow \left\{ \matrix{
{9^x} - 6 > 0 \hfill \cr
{4.3^x} - 6 = 2\left( {{9^x} - 6} \right) \hfill \cr} \right. \Leftrightarrow \left\{ \matrix{
t > \sqrt 6 \hfill \cr
2{t^2} - 4t - 6 = 0 \hfill \cr} \right.\) (với \(t = {3^x}\))
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\( \Leftrightarrow t = 3 \Leftrightarrow {3^x} = 3 \Leftrightarrow x = 1\)
Vậy \(S = \left\{ 1 \right\}\)
c) Điều kiện: \(x >9\)
\(\eqalign{
& 1 - {1 \over 2}\log \left( {2x - 1} \right) = {1 \over 2}\log \left( {x - 9} \right) \Leftrightarrow 2 = \log \left( {2x - 1} \right) + \log \left( {x - 9} \right) \cr
& \Leftrightarrow \log \left( {2x - 1} \right)\left( {x - 9} \right) = 2 \Leftrightarrow \left( {2x - 1} \right)\left( {x - 9} \right) = 100 \cr
& \Leftrightarrow 2{x^2} - 19x - 91 = 0 \Leftrightarrow \left[ \matrix{
x = 13 \hfill \cr
x = - 3,5\,\,\left( \text {loại} \right) \hfill \cr} \right. \cr} \)
Vậy \(x=13\)
d) Điều kiện: \(x > 2\)
Ta có: \({\log _{{1 \over 8}}}\sqrt {3x - 5} = {\log _{{2^{ - 3}}}}{\left( {3x - 5} \right)^{{1 \over 2}}} = - {1 \over 6}{\log _2}\left( {3x - 5} \right)\)
Phương trình đã có trở thành:
\(\eqalign{
& {1 \over 6}{\log _2}\left( {x - 2} \right) + {1 \over 6}{\log _2}\left( {3x - 5} \right) = {1 \over 3} \cr
& \Leftrightarrow {\log _2}\left( {x - 2} \right)\left( {3x - 5} \right) = 2 \cr
& \Leftrightarrow \left( {x - 2} \right)\left( {3x - 5} \right) = 4 \cr
& \Leftrightarrow x = 3\,\,\text{ hoặc }\,\,x = {2 \over 3}. \cr} \)
Với điều kiện \(x > 2\) ta chỉ nhận nghiệm \(x = 3\).
Vậy \(S = \left\{ 3 \right\}\)