Logarithmic Equations and Inequalities

If ${\log }_{x}a, a^x$ and ${\log }_{b}x$ are in $GP$, then what is $x$ equal to?

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

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

Let $\frac{{\log }_x\left(2a-x\right)}{{\log }_x\left(2\right)}+\frac{{\log }_a\left(x\right)}{{\log }_a\left(2\right)}=\frac{1}{{\log }_{a^2-1}\left(2\right)}$, then sum of possible value(s) of '$a$' for which equation has only one solution will lie in interval

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

The number of solution(s) of the equation ${\log }_2\left(\frac{x-1}{x+5}\right)+{\log }_2\left(x^2+6x+5\right)=3$, is

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

If $\log 3^{x+4}=\log 729$ then the value of $x$ will be

The value for the given  ${\log }_{\frac{9}{4}}\left(\frac{1}{2\sqrt{3}}\sqrt{6-\frac{1}{2\sqrt{3}}\sqrt{6-\frac{1}{2\sqrt{3}}\sqrt{6-\frac{1}{2\sqrt{3}}\dots ..\infty }}}\right)$

The given numbers of real solution of the equation $\sqrt{{\log }_{10}\left(-x\right)}={\log }_{10}\sqrt{x^2}$ is$.........$

If ${\log }_e\left(y^2\right)=\tan \left(\frac{1}{x}\right),$ then $y=$

Find the value of $n$ for the equation ${\log }_{2}3{\log }_{2}4{\log }_{2}5.......{\log }_2\left(n+1\right)=10$.

If ${\log }_{8}x+{\log }_{4}x+{\log }_{2}x=11,x=$ ?

Find the number of value of $x$: ${\log }_9(x+1).{\log }_2\left(x+1\right)-{\log }_9\left(x+1\right)-{\log }_2\left(x+1\right)+1<0$

The value of $x$ that satisfies $2{\log }_{\frac{1}{4}}(x+5)>\frac{9}{4}{\log }_{\frac{1}{3\sqrt{3}}}\left(9\right)+{\log }_{\sqrt{x+5}} \left(2\right)$

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

The least value of the quantity $2 {\log }_{10}\left(x\right)-{\log }_x\left(0.01\right) \left[x>1\right]=$

If $a={\log }_{3}5, b={\log }_{17}25$, then show that $a>b$.

Prove that $\frac{1}{{\log }_2\mathrm{\pi }}+\frac{1}{{\log }_6\mathrm{\pi }}>2$

If ${\log }_{\left(9y+25\right)}\left(1849\right)=2$, the value of $y$ is _____