How to solve logarithmic inequality ?

• Example 1 Solve the inequality \(\log x \leq 3\)
  Solution - \begin{eqnarray}\log x &\leq& 3 \\\nonumber x &\leq& 10^3 \\\nonumber x &\leq& 1000 \\\nonumber x&\in& \Big\langle -\infty, 1000\Big]\end{eqnarray}
• Example 2 Solve the inequality \(\log_x 25 \geq 2 \)
  Solution - In this inequality two cases are considered one where the logarithmic base \(x\) is in range between 0 and 1 and the other case where the logarithmic base \(x\) is greater than 1. Case I \(0 < x < 1 \)\begin{eqnarray}\log_x 25 &\geq& 2 \\\nonumber x^2 &\geq& 5^2 \\\nonumber |x| &\geq& 5 \Rightarrow x \leq -5 \mathrm{or} x \geq 5. \end{eqnarray} Since in this case the value of logarithm base is between 0 and 1 this case do not have solution. Case 2 \(x >1\) \begin{eqnarray} \log_x 25 &\geq& 2 \\\nonumber |x| &\leq& 5 \Rightarrow -5 \leq x \leq 5 \end{eqnarray} In this case the solution of inequality is \(1 < x \leq 5.\)
• Example 3 Solve the inequality \(\log_5 x > 3\)
  Solution - \begin{eqnarray}\log_5 x &>& 3 \\\nonumber x &>& 5^3\\\nonumber x &>& 125 \\\nonumber x &\in& \Big\langle 125, +\infty\Big\rangle\end{eqnarray}
• Example 4 Solve the inequality \(\log_{0.2} (x-2) > -2\)
  Solution - \begin{eqnarray}\log_{0.2} (x-2) &>& -2\\\nonumber \log_\frac{2}{10} (x-2) &>& -2 \\\nonumber -\log_5(x-2) &>& -2 /\cdot(-1) \\\nonumber \log_5(x-2) &<& 2\\\nonumber x-2 &<& 5^2 \\\nonumber x &<& 27\end{eqnarray}
• Example 5 Solve the inequality \(\frac{1}{\log_x -1} < \frac{1}{1+\log_x}\)
  Solution - \begin{eqnarray}\frac{1}{\log_x -1} &<& \frac{1}{1+\log_x} \Rightarrow log_x = t \\\nonumber \frac{1}{t-1} &<& \frac{1}{1+t}\\\nonumber \frac{1}{t-1}-\frac{1}{1+t} &<& 0 \\\nonumber \frac{1+t-t+1}{(1+t)(t-1)}\\\nonumber t-1 &<& 0 \Rightarrow t < 1 \\\nonumber 1+t &<& 0 \Rightarrow t &<& -1 \\\nonumber \log x_1 &<& 1 \Rightarrow x_1 &<& 10 \\\nonumber \log x_2 &<& -1 \Rightarrow x_2 &<& \frac{1}{10} \\\nonumber x &\in& \Big\langle \frac{1}{10},10\Big\rangle\end{eqnarray}
• Example 6 Solve the inequaltiy \(\log_{0.5} x > -3 \)
  Solution - \begin{eqnarray}\log_{0.5} x &>& -3\\\nonumber -\log_2 x &>& -3 /\cdot(-1)\\\nonumber \log_2 x &<& 3\\\nonumber x &<& 2^3 \\\nonumber x &<& 8 \Rightarrow x\in \Big\langle -\infty, 8 \Big\rangle \end{eqnarray}
• Example 7 Solve the inequality \(\frac{\log x + 1}{\log x} < 0 \)
  Solution - \begin{eqnarray}\frac{\log x + 1}{\log x} &<& 0 \\\nonumber \log_x +1 &<& 0 \Rightarrow \log_x <-1 \Rightarrow x < \frac{1}{10} \\\nonumber \log_x &<& 0\Rightarrow x < 1 \\\nonumber x &\in& \Big\langle \frac{1}{10},1\Big\rangle \end{eqnarray}
• Example 8 Solve the inequality \(\log_2(2x-1) \leq 4 \)
  Solution - \begin{eqnarray}\log_2(2x-1) &\leq& 4\\\nonumber 2x-1 &\leq& 2^4 \\\nonumber 2x &\leq& 17/:2 \\\nonumber x &\leq& \frac{17}{2} \Rightarrow x &\in& \Big[ 0, \frac{17}{2}\Big] \end{eqnarray}
• Example 9 Solve the inequality \(\log_4 (3-2x) \geq \frac{1}{2}\)
  Solution - \begin{eqnarray}\log_4 (3-2x) &\geq& \frac{1}{2}\\\nonumber 3-2x &\geq& 2 \\\nonumber -2x &\geq& -1/:(-2) \\\nonumber x &\geq& \frac{1}{2} \Rightarrow x &\in& \Big\langle -\infty , \frac{1}{2}\Big]\end{eqnarray}
• Example 10 Solve the inequality \(\log_\frac{1}{3} < 3 \)
  Solution - \begin{eqnarray}\log_\frac{1}{3} &<& 3\\\nonumber -\log_3 x &<& 3 /\cdot(-1) \\\nonumber x &>& 3^{-3} \Rightarrow x &>& \frac{1}{27} \end{eqnarray}
• Example 11 Solve the inequality \(\log(-2x) \geq -1\)
  Solution - \begin{eqnarray}\log(-2x) &\geq& -1 \\\nonumber -2x &\geq& \frac{1}{10}/:(-2) \\\nonumber x &\leq& -\frac{1}{20}\end{eqnarray}
• Example 12 Solve the inequaltiy \(\log_3(2x-1) \leq 4\)
  Solution - \begin{eqnarray}\log_3(2x-1) &\leq& 4\\\nonumber 2x-1 &\leq& 81 \\\nonumber 2x &\leq& 82 /:2 \\\nonumber x &\leq& 41\end{eqnarray} Argument of logaritmic function must be greater than 0 $$ 2x - 1 > 0 \Rightarrow 2x > 1 /: 2 \Rightarrow x > \frac{1}{2}$$ The solution of inequality can be written as \(x\in\Big\langle\frac{1}{2}, 41\Big]\)
• Example 13 Solve the inequality \(\ln(2x-3) < 5\)
  Solution - \begin{eqnarray}\ln(2x-3) &<& 5 \\\nonumber 2x-3 &<& e^5 \\\nonumber 2x &<& e^5 +3/:2 \\\nonumber x &<& \frac{e^5+3}{2}\end{eqnarray} Arugment of natural logarithm function must be strictly greater than 0. $$2x-3 >0 \Rightarrow 2x > 3 /:2 \Rightarrow x > \frac{3}{2} $$ The solution of inequality \(x\in \Big\langle\frac{3}{2}, \frac{e^5 +3}{2} \Big\rangle\)
• Example 14 Solve the inequality \(\log_{0.2} (3x) > -3\)
  Solution - \begin{eqnarray}\log_{0.2} (3x) &>& -3\\\nonumber \log_\frac{1}{5}(3x) &>& -3 \\\nonumber -\log_5 (3x) &>& -3 /\cdot(-1) \\\nonumber \log_5(3x) &<& 3 \\\nonumber 3x &<& 5^3 \\\nonumber x < \frac{125}{3} \end{eqnarray} The argument of logartihm function must be strictly greater than 0. $$3x > 0 \Rightarrow x > 0 $$ The solution of inequality can be written as \(x \in \Big\langle 0, \frac{125}{3}\Big\rangle\)
• Example 15 Solve the inequality \(\ln(4-7x) > -3\)
  Solution - \begin{eqnarray}\ln(4-7x) &>& -3\\\nonumber 4-7x &>& e^{-3} \\\nonumber -7x &>& e^{-3} -4 /\cdot(-1)/:7 \\\nonumber x< \frac{1-4e^3}{7e^3}\end{eqnarray}
• Example 16 Solve the inequality \(\log_x 6 \geq 4\)
  Solution - \begin{eqnarray}\log_x 6 &\geq& 4 \\\nonumber x &\leq& \sqrt[4]{6}\end{eqnarray} The base of the logarithm must be greater than 1. The entire solution of the inequality is \(x\in \Big\langle 1, \sqrt[4]{6} \Big]\)
• Example 17 Solve the inequality \(\log_x 8 < -3\)
  Solution - \begin{eqnarray}\log_x 8 &<& -3\\\nonumber 2^3 &<& x^-3 \\\nonumber \frac{1}{x^3} < 8 \\\nonumber x^3 &>& \frac{1}{8}\end{eqnarray} The solution of the inequality can be written as \(x \in \langle 0, \frac{1}{2}\rangle \cup \langle 1, +\infty\rangle\)
• Example 18 Solve the inequality \(\log 10^{7-2x} < 6\)
  Solution - \begin{eqnarray}\log 10^{7-2x} &<& 6 \\\nonumber 10^{7-2x} &<& 10^6 \\\nonumber 7-2x &<& 6 \\\nonumber -2x&<& -1 \\\nonumber x &>& \frac{1}{2}\end{eqnarray}
• Example 19 Solve the inequality \(\log_{\frac{1}{3}} \frac{x-3}{x} > -1\)
  Solution - \begin{eqnarray}\log_{\frac{1}{3}} \frac{x-3}{x} &>& -1 \\\nonumber -\log_3 \frac{x-3}{x} &>& -1/\cdot(-1)\\\nonumber \log_3 \frac{x-3}{x} &<& 1 \\\nonumber \frac{x-3}{x} &<& 3 \\\nonumber x-3 &<& 3x \\\nonumber x-3x &<& 3 \\\nonumber -2x &<& 3/:(-2) \\\nonumber x &>& -\frac{3}{2}\end{eqnarray} The argument of logarithm function ???
• Example 20 Solve the inequality \(\frac{-2x-x^2}{\log(-x)}\geq 0\)
  Solution - \begin{eqnarray}\frac{-2x-x^2}{\log(-x)}&\geq& 0 \\\nonumber \log(-x) &\geq& 0 \\\nonumber -x &>& 10^0 \Rightarrow x &<&-1 \\\nonumber -2x -x^2 &\geq& 0 \\\nonumber 2x+x^2 &\leq& 0 \\\nonumber x_1 &=& 0, x_2 = -2\\\nonumber x&\in& \Big[-2,-1\Big\rangle\end{eqnarray}