Logarithmic Function and its Properties

If ${\log }_3\left(23+4{\log }_3\left(x-1\right)\right)=3$ then $x$ is

The number of integral solution $x$ of ${\log }_{\left(x+\frac{7}{2}\right)}{\left(\frac{x-7}{2x-3}\right)}^2\ge 0$ is

Solution of inequality

${\log }_{\sin x}\left({\log }_3\left({\log }_{0.2}\left(x\right)\right)\right)<0$ is

Let $x,y$ and $z$ be real numbers satisfying the system 

${\log }_2\left(xyz-3+{\log }_5\left(x\right)\right)=5$

${\log }_3\left(xyz-3+{\log }_5\left(y\right)\right)=4$

${\log }_4\left(xyz-3+{\log }_5\left(z\right)\right)=4$

then, the value of ${\log }_{5}x+{\log }_{5}y+{\log }_{5}z+3xyz$ is

If ${\log }_3\left[{\log }_3\sqrt{x-1}-\sqrt{x}\right]=0$ then

If ${\log }_3\left({\log }_2\left(a\right)\right)+{\log }_{\frac{1}{3}}\left({\log }_{\frac{1}{2}}\left(b\right)\right)=1$, then the value of $a{b}^3$ is

Values of ${\log }_{\left(\sqrt{3}-\sqrt{2}\right)}\left(5-2\sqrt{6}\right)$ is

${\log }_{0.5}{x}^2>{\log }_{0.5}25,$ then the possible value of $x$ is

$\log \left[\frac{\sqrt[3]{a^{-1}b{c}^{-2}}}{{\left(a^{-2}{b}^{-1}{c}^{-4}\right)}^{{\displaystyle \raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$6$}\right.}}}\right]=$

If ${\log }_{10}0.5=1.6990$, then number of zeros that are between the decimal point and the first significat figure in ${\left(0.5\right)}^{100}$ is $k$, then $k$ is divisible by

The value of ${10}^{\left({\log }_{10}2+{\log }_{10}3+\dots +{\log }_{10}n\right)}$ is

If $\left({\log }_{a}2\right)\sin \theta =\left({\log }_{b}2\right)\sin \left(\theta +\frac{2\pi }{3}\right)=\left({\log }_{c}2\right)\sin \left(\theta +\frac{4\pi }{3}\right)$ (wherever defined), then $abc$ is equal to-

$\frac{{\log }_{e}a}{y-z}=\frac{{\log }_{e}b}{z-x}=\frac{{\log }_{e}c}{x-y}$, then $a^{y^2+yz+z^2}{b}^{z^2+zx+x^2} c^{x^2+xy+y^2}$ is equal to:

${\log }_{\left(x^2+1\right)}\left(5-x\right)$ is defined for all $x\in$

If $\frac{a+{\log }_{4}3}{a+{\log }_{2}3}=\frac{a+{\log }_{8}3}{a+{\log }_{4}3}=b$, then $b=$

$a,b,c$ are real numbers greater then $1$ and $x\in R^{+}$. If ${\log }_{a}x=24,{\log }_{b}x=40,{\log }_{abc}x=12$ then ${\log }_{c}x$ equal is

Solve for $x$: $2+\log x=\log 500$

If $\frac{\log x}{a^2+ab+b^2}=\frac{\log y}{b^2+bc+c^2}=\frac{\log z}{c^2+ca+a^2}$, then

$x^{a-b}{y}^{b-c}{z}^{c-a}=$

If $\frac{\ln a}{y-z}=\frac{\ln b}{z-x}=\frac{\ln c}{x-y}$, then

$a^{y^2+yz+z^2}{b}^{z^2+zx+x^2}{c}^{x^2+xy+y^2}=$