Logarithmic Equations and 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 

If ${\log }_{2}x=\frac{1}{3}.{\log }_{2}8-2{\log }_2\sqrt{3}$, then $x$ is equal to

If ${\mathrm{l}\mathrm{o}\mathrm{g}}_e{\mathrm{l}\mathrm{o}\mathrm{g}}_5\left[\sqrt{2x-2}+3\right]=0$ then the value of $x$ is-

The number of solution$\left(s\right)$ of the equation, $\log \left(-2x\right)=\log x^2$ is

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

Number of real values of $x$ which satisfy $\mathrm{l}\mathrm{o}{\mathrm{g}}_2{x}^2+{\log }_2\left(x+2\right)=4$ is/are

Let $f\left(x\right)=\log \mathrm{ }\left(\frac{2\mathrm{x}-1}{2{\mathrm{x}}^3+3{\mathrm{x}}^2+\mathrm{x}}\right)$, then domain of $f\left(x\right)$ contain the set

If $1-{\mathrm{l}\mathrm{o}\mathrm{g}}_{10}5=\frac{1}{3}\left({\log }_{10}\frac{1}{2}+{\log }_{10}x+\frac{1}{3}{\log }_{10}5\right)$ then $x$ is equal to

A controlled experiment took place in a laboratory. Biologists estimated the number of microbes $N$ present in a culture $t$ days after the start of their experiment. The first results obtained are shown in the top two rows of the table below.

$\begin{array}{|lccccc|}\hline \mathit{t}& 3& 5& 10& 15& 20\\ \mathit{N}& 115& 178& 480& 900& 1600\\ \log N& & & & & \\ \hline\end{array}$

Complete the last row of the table, giving values to $2$ decimal places.

Draw a graph of $\log N$ against $t$, using appropriate  scales.

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

${\log }_8+{\log }_8\left(x^2+\mathrm{x}-2\right)=\frac{4}{3} ,$ then$x$ is equal to-

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 $\frac{2}{5}$.

Let $x, y$ be real numbers such that $x>2y>0$ and $2\log \left(x-2y\right)=\log x+\log y$. Then the possible value(s) of $\frac{x}{y}$

Find the domain of the function $f\left(x\right)={\log }_4(4x-3)$.

Find the domain of the function $f\left(x\right)={\log }_4(2x-3)$.

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