Logarithmic Inequalities

Solution of the inequality ${\log }_{5-x}\left(3x-1\right)\ge {\log }_{5-x}{\left(x+1\right)}^2$ is 

Find the solution set of the inequality ${\log }_{2x+3}\left(x^2\right)<{\log }_{2x+3}\left(2x+3\right)$.

Number of integral values of $x$ the inequality ${\log }_{10}\left(\frac{2x-2007}{x+1}\right)\le 0$ holds true, is

$\left\{x:{\log }_{\frac{1}{3}}\left({\log }_4\left(x^2-5\right)\right)>0\right\}=$ ______.

The solution set of ${\log }_{\left|\sin x\right|}\left(x^2-8x+23\right)>\frac{3}{{\log }_2\left|\sin x\right|}$ contains

Let $ABC$ is a triangle. $AB={\log }_e\left(8\right), BC={\log }_e\left(50\right)$ and $AC={\log }_e\left(n\right)$ where $n$ is a positive integer. Then the largest possible value of $n$ is

The set of numeric values of $x$ which satisfy the equation $\mathrm{l}\mathrm{o}{\mathrm{g}}_{x^2}\left(x^2-1\right)\le 0$ is

Consider the inequality ${\log }_x\left(\frac{15}{1-2x}\right)<-2.$ Which of the following is/are CORRECT?

The complete solution set of the inequality $\frac{1}{{\log }_4\left(\frac{x+1}{x+2}\right)}>\frac{1}{{\log }_4\left(x+3\right)}$ is $\left(\frac{-a}{20},\infty \right)$, then determine $a$

Which of the following intervals contain complete solution set of the inequality ${\log }_{1/5}\left({\log }_4\left(x^2-5\right)\right)\ge 0$ ?

Which of the following are true?

The least integral value of $x$ satisfying the inequality ${\log }_{\frac{1}{5}}\left(x^2-6x+18\right)+2{\log }_5\left(x-4\right)<0$ is $\lambda$ then the value of  $3\lambda$ is

If $\alpha$ is one of the solution of inequation ${\left({\log }_{10}100x\right)}^2+{\left({\log }_{10}10x\right)}^2+{\log }_{10}x\le 14$ then $\alpha$ can belongs to

The inequality $\frac{1}{{\log }_{2}x}-\frac{1}{{\displaystyle {\log }_{2}x -1}}<1$ is satisfied by 

The possible values of $a$ for which any solution of the inequality, $\frac{{\log }_3\left(x^2-3x+7\right)}{{\log }_3(3x+2)}<1$ is also a solution of the inequality $x^2+(5-2a)x\le 10a$ can be

If the complete solution set of the inequality ${\left({\log }_{10}x\right)}^2\ge {\log }_{10}x+2$ is $(0,a]\cup [b,\infty )$ then :

If the complete solution set of the inequation ${\log }_{1/2}{\log }_6\left(\frac{x^2+x}{x+4}\right)<0\text{ is ( }a,b)\cup (c,\infty )$, then find the value of $c-a-b$

If the set of all values of parameter $c$ for which the inequality $1+{\log }_2\left(2x^2+2x+\frac{7}{2}\right)\ge {\log }_2\left(c{x}^2+c\right)$
possesses at least one solution is $S$ then $S$ contained in

If $\frac{{\log }_5\left(x^2-5x+7\right)}{{\log }_5(0.001)}>0,$ then $x\in (a,b).$ Find $5 b-a$

If the solution set of inequation $\frac{1-{\log }_{1/2}(-x)}{\sqrt{-2-6x}}<0$ is $(a, b)$ then $-$