Hãy nhập câu hỏi của bạn vào đây, nếu là tài khoản VIP, bạn sẽ được ưu tiên trả lời.
\(a,\left(0,3\right)^{x-3}=1\\ \Leftrightarrow x-3=0\\ \Leftrightarrow x=3\\ b,5^{3x-2}=25\\ \Leftrightarrow3x-2=2\\ \Leftrightarrow3x=4\\ \Leftrightarrow x=\dfrac{4}{3}\\ c,9^{x-2}=243^{x+1}\\ \Leftrightarrow3^{2x-4}=3^{5x+5}\\ \Leftrightarrow2x-4=5x+5\\ \Leftrightarrow3x=-9\\ \Leftrightarrow x=-3\)
d, Điều kiện: \(x>-1;x\ne0\)
\(log_{\dfrac{1}{x}}\left(x+1\right)=-3\\ \Leftrightarrow x+1=x^3\\ x\simeq1,325\left(tm\right)\)
e, Điều kiện: \(x>\dfrac{5}{3}\)
\(log_5\left(3x-5\right)=log_5\left(2x+1\right)\\ \Leftrightarrow3x-5=2x+1\\ \Leftrightarrow x=6\left(tm\right)\)
f, Điều kiện: \(x>\dfrac{1}{2}\)
\(log_{\dfrac{1}{7}}\left(x+9\right)=log_{\dfrac{1}{7}}\left(2x-1\right)\\ \Leftrightarrow x+9=2x-1\\ \Leftrightarrow x=10\left(tm\right)\)
a) \({3^{{x^2} - 4x + 5}} = 9 \Leftrightarrow {x^2} - 4x + 5 = 2 \Leftrightarrow {x^2} - 4x + 3 = 0 \Leftrightarrow \left( {x - 3} \right)\left( {x - 1} \right) = 0\)
\( \Leftrightarrow \left[ \begin{array}{l}x = 3\\x = 1\end{array} \right.\)
Vậy phương trình có nghiệm là \(x \in \left\{ {1;3} \right\}\)
b) \(0,{5^{2x - 4}} = 4 \Leftrightarrow 2x - 4 = {\log _{0,5}}4 \Leftrightarrow 2x = 2 \Leftrightarrow x = 1\)
Vậy phương trình có nghiệm là x = 1
c) \({\log _3}(2x - 1) = 3\) ĐK: \(2x - 1 > 0 \Leftrightarrow x > \frac{1}{2}\)
\( \Leftrightarrow 2x - 1 = 27 \Leftrightarrow x = 14\) (TMĐK)
Vậy phương trình có nghiệm là x = 14
d) \(\log x + \log (x - 3) = 1\) ĐK: \(x - 3 > 0 \Leftrightarrow x > 3\)
\(\begin{array}{l} \Leftrightarrow \log \left( {x.\left( {x - 3} \right)} \right) = 1\\ \Leftrightarrow {x^2} - 3x = 10\\ \Leftrightarrow {x^2} - 3x - 10 = 0\\ \Leftrightarrow \left( {x + 2} \right)\left( {x - 5} \right) = 0\\ \Leftrightarrow \left[ \begin{array}{l}x = - 2 (loại) \,\,\,\\x = 5 (TMĐK) \,\,\,\,\,\,\,\end{array} \right.\end{array}\)
Vậy phương trình có nghiệm x = 5
\(a,3^x>\dfrac{1}{243}\\ \Leftrightarrow3^x>3^{-5}\\ \Leftrightarrow x>-5\\ b,\left(\dfrac{2}{3}\right)^{3x-7}\le\dfrac{3}{2}\\ \Leftrightarrow3x-7\le1\\ \Leftrightarrow3x\le8\\ \Leftrightarrow x\le\dfrac{8}{3}\\ c,4^{x+3}\ge32^x\\ \Leftrightarrow2^{2x+6}\ge2^{5x}\\ \Leftrightarrow2x+6\ge5x\\ \Leftrightarrow3x\le6\\ \Leftrightarrow x\le2\)
d, Điều kiện: x > 1
\(log\left(x-1\right)< 0\\ \Leftrightarrow x-1< 1\\ \Leftrightarrow1< x< 2\)
e, Điều kiện: \(x>\dfrac{1}{2}\)
\(log_{\dfrac{1}{5}}\left(2x-1\right)\ge log_{\dfrac{1}{5}}\left(x+3\right)\\ \Leftrightarrow2x-1\ge x+3\\ \Leftrightarrow x\ge4\)
f, Điều kiện: x > 4
\(ln\left(x+3\right)\ge ln\left(2x-8\right)\\ \Leftrightarrow x+3\ge2x-8\\\Leftrightarrow4< x\le11\)
a) \({\log _{\frac{1}{7}}}\left( {x + 1} \right) > {\log _7}\left( {2 - x} \right)\) (ĐK: \(x + 1 > 0;2 - x > 0 \Leftrightarrow - 1 < x < 2\))
\(\begin{array}{l} \Leftrightarrow {\log _{{7^{ - 1}}}}\left( {x + 1} \right) > {\log _7}\left( {2 - x} \right)\\ \Leftrightarrow - {\log _7}\left( {x + 1} \right) > {\log _7}\left( {2 - x} \right)\\ \Leftrightarrow {\log _7}{\left( {x + 1} \right)^{ - 1}} > {\log _7}\left( {2 - x} \right)\\ \Leftrightarrow {\left( {x + 1} \right)^{ - 1}} > 2 - x\\ \Leftrightarrow \frac{1}{{x + 1}} - 2 + x > 0\\ \Leftrightarrow \frac{{1 + \left( {x - 2} \right)\left( {x + 1} \right)}}{{x + 1}} > 0\\ \Leftrightarrow \frac{{1 + {x^2} - x - 2}}{{x + 1}} > 0 \Leftrightarrow \frac{{{x^2} - x - 1}}{{x + 1}} > 0\end{array}\)
Mà – 1 < x < 2 nên x + 1 > 0
\( \Leftrightarrow {x^2} - x - 1 > 0 \Leftrightarrow \left[ \begin{array}{l}x < \frac{{1 - \sqrt 5 }}{2}\\x > \frac{{1 + \sqrt 5 }}{2}\end{array} \right.\)
KHĐK ta có \(\left[ \begin{array}{l} - 1 < x < \frac{{1 - \sqrt 5 }}{2}\\\frac{{1 + \sqrt 5 }}{2} < x < 2\end{array} \right.\)
b) \(2\log \left( {2x + 1} \right) > 3\) (ĐK: \(2x + 1 > 0 \Leftrightarrow x > \frac{{ - 1}}{2}\))
\(\begin{array}{l} \Leftrightarrow \log \left( {2x + 1} \right) > \frac{3}{2}\\ \Leftrightarrow 2x + 1 > {10^{\frac{3}{2}}} = 10\sqrt {10} \\ \Leftrightarrow x > \frac{{10\sqrt {10} - 1}}{2}\end{array}\)
KHĐK ta có \(x > \frac{{10\sqrt {10} - 1}}{2}\)
a)
ĐK: \(\left\{{}\begin{matrix}2x-4>0\\x-1>0\end{matrix}\right.\Leftrightarrow x>1\)
\(\log_5\left(2x-4\right)+\log_{\dfrac{1}{5}}\left(x-1\right)=0\\ \Leftrightarrow\log_5\left(2x-4\right)-\log_5\left(x-1\right)=0\\ \Leftrightarrow\log_5\left(\dfrac{2x-4}{x-1}\right)=\log_51\\ \Leftrightarrow\dfrac{2x-4}{x-1}=1\\ \Leftrightarrow2x-4=x-1\\ \Leftrightarrow x=3\left(tm\right)\)
Vậy x = 3.
b) ĐK: x > 0
\(\log_2x+\log_4x=3\\ \Leftrightarrow\log_2x+\dfrac{1}{2}\log_2x=3\\ \Leftrightarrow\left(1+\dfrac{1}{2}\right)\log_2x=3\\ \Leftrightarrow\dfrac{3}{2}\log_2x=3\\ \Leftrightarrow\log_2x=2\\ \Leftrightarrow x=4\left(tm\right)\)
Vậy x= 4
\(a,0,1^{2-x}>0,1^{4+2x}\\ \Leftrightarrow2-x>2x+4\\ \Leftrightarrow3x< -2\\ \Leftrightarrow x< -\dfrac{2}{3}\)
\(b,2\cdot5^{2x+1}\le3\\ \Leftrightarrow5^{2x+1}\le\dfrac{3}{2}\\ \Leftrightarrow2x+1\le log_5\left(\dfrac{3}{2}\right)\\ \Leftrightarrow2x\le log_5\left(\dfrac{3}{2}\right)-1\\ \Leftrightarrow x\le\dfrac{1}{2}log_5\left(\dfrac{3}{2}\right)-\dfrac{1}{2}\\ \Leftrightarrow x\le log_5\left(\dfrac{\sqrt{30}}{10}\right)\)
c, ĐK: \(x>-7\)
\(log_3\left(x+7\right)\ge-1\\ \Leftrightarrow x+7\ge\dfrac{1}{3}\\ \Leftrightarrow x\ge-\dfrac{20}{3}\)
Kết hợp với ĐKXĐ, ta có:\(x\ge-\dfrac{20}{3}\)
d, ĐK: \(x>\dfrac{1}{2}\)
\(log_{0,5}\left(x+7\right)\ge log_{0,5}\left(2x-1\right)\\ \Leftrightarrow x+7\le2x-1\\ \Leftrightarrow x\ge8\)
Kết hợp với ĐKXĐ, ta được: \(x\ge8\)
a, Điều kiện: x > 0
\(log_3\left(x\right)< 2\\ \Rightarrow0< x< 9\)
b, Điều kiện: x > 5
\(log_{\dfrac{1}{4}}\left(x-5\right)\ge-2\\ \Rightarrow x-5\le16\\ \Leftrightarrow5< x\le21\)
a, ĐK: \(x+1>0\Leftrightarrow x>-1\)
\(log\left(x+1\right)=2\\ \Leftrightarrow x+1=10^2\\ \Leftrightarrow x+1=100\\ \Leftrightarrow x=99\left(tm\right)\)
b, ĐK: \(\left\{{}\begin{matrix}x-3>0\\x>0\end{matrix}\right.\Rightarrow x>3\)
\(2log_4x+log_2\left(x-3\right)=2\\ \Leftrightarrow log_2x+log_2\left(x-3\right)=2\\ \Leftrightarrow log_2\left(x^2-3x\right)=2\\ \Leftrightarrow x^2-3x=4\\ \Leftrightarrow x^2-3x-4=0\\ \Leftrightarrow\left(x+1\right)\left(x-4\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=-1\left(ktm\right)\\x=4\left(tm\right)\end{matrix}\right.\)
c, ĐK: \(x>1\)
\(lnx+ln\left(x-1\right)=ln4x\\ \Leftrightarrow ln\left[x\left(x-1\right)\right]-ln4x=0\\ \Leftrightarrow ln\left(\dfrac{x-1}{4}\right)=0\\ \Leftrightarrow\dfrac{x-1}{4}=1\\ \Leftrightarrow x-1=4\\ \Leftrightarrow x=5\left(tm\right)\)
d, ĐK: \(\left\{{}\begin{matrix}x^2-3x+2>0\\2x-4>0\end{matrix}\right.\Rightarrow x>2\)
\(log_3\left(x^2-3x+2\right)=log_3\left(2x-4\right)\\ \Leftrightarrow x^2-3x+2=2x-4\\ \Leftrightarrow x^2-5x+6=0\\ \Leftrightarrow\left(x-2\right)\left(x-3\right)=0\\ \Leftrightarrow\left[{}\begin{matrix}x=2\left(ktm\right)\\x=3\left(tm\right)\end{matrix}\right.\)
