Logarithmic Inequalities

Let $n$ be the smallest positive integer such that $1+\frac{1}{2}+\frac{1}{3}+\dots +\frac{1}{n}\ge 4$. Which one of the following statements is true?

The number of integers satisfying the inequality ${\log }_{\sqrt{0.9}} {\log }_5\left(\sqrt{x^2+5+x}\right)>0$ is:

Number of integral solutions of inequality ${\left(\frac{1}{10}\right)}^{{\log }_{\left(\mathrm{x}-3\right)}\left({\mathrm{x}}^2-4\mathrm{x}+3\right)}$$\ge 1$ are 

Number of integers satisfying inequality, $\sqrt{{\log }_3\left(x\right)-1}+\frac{\frac{1}{2}{\log }_3{x}^3}{{\log }_3\left(\frac{1}{3}\right)}+2>0$ is

Solve ${\log }_3\left(5x-1\right)>0$

Solution of inequality ${\log }_{{\log }_2\left(\frac{x}{2}\right)}\left(x^2-10x+22\right)>0$ is

Prove that the value of ${\log }_{10}3$ lies in between $\frac{1}{2}$ and $\frac{2}{5}$.

Prove that the value of ${\log }_{10}3$ lies in between $\frac{1}{2}$ and $\frac{2}{5}$.

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