Properties of Logarithms

If $N=m!$ (where $m$ is a fixed positive integer $>2$), then $\frac{1}{{\log }_{2}N}+\frac{1}{{\log }_{3}N}+\frac{1}{{\log }_{4}N}+\mathrm{.....}+\frac{1}{{\log }_{m}N}=$

${\log }_7{\log }_7\sqrt{7\sqrt{7\sqrt{7}}}$ is equal to

If $2^x{.3}^{x+4}=7^x$, then $x$ is equal to

Let $a,b\in R^{+}$ for which ${\left(60\right)}^a=3$ and ${\left(60\right)}^b=5,$ then ${12}^{\frac{1-a-b}{2\left(1-b\right)}}$ is equal to

 a, b, c are positive real numbers such that  $a{}^{{\log }_{3}7}=27;$ $b^{{\log }_{7}11}=49$ and $c^{{\log }_{11}25}=\sqrt{11}$. The value of  $\left(a^{{\left({\log }_{3}7\right)}^2}+b^{{\left({\log }_{7}11\right)}^2}+c^{{\left({\log }_{11}25\right)}^2}\right)$  equals

The sum of the series $\frac{1}{{\log }_{2}4}+\frac{1}{{\log }_{4}4}+\frac{1}{{\log }_{8}4}+\dots \dots .+\frac{1}{{\log }_{2^n}4}$ is $\alpha ,$ then $\frac{\alpha }{(1+2+3+\dots +n)}$ is

Let ${\log }_{10}2=a \& {\log }_{10}3=b$, then ${\log }_{\sqrt{\mathrm{45}}}\mathrm{144}$ is.

The value of $5^{{\log }_{\sqrt{5}}3}+9^{{\log }_{3}6}-8^{{\log }_{2}7}$ is.

$a^{\frac{\mathrm{l}\mathrm{o}{\mathrm{g}}_b\left({\log }_{\mathrm{b}}\mathrm{N}\right)}{\mathrm{l}\mathrm{o}{\mathrm{g}}_{b}a}}$, is equal to

If $a=\mathrm{l}\mathrm{o}{\mathrm{g}}_{2}3, b=\mathrm{l}\mathrm{o}{\mathrm{g}}_{3}5, c=\mathrm{l}\mathrm{o}{\mathrm{g}}_{7}2$, then$\hspace{0.17em}{\log }_{140}63$, in terms of $a, b \& c$, is.

If $\mathrm{l}\mathrm{o}{\mathrm{g}}_{10}2=a$ and $\mathrm{l}\mathrm{o}{\mathrm{g}}_{10}6=b$ then $\frac{\mathrm{l}\mathrm{o}{\mathrm{g}}_{10}80}{\mathrm{l}\mathrm{o}{\mathrm{g}}_{10}12}$ is equal to-

${\mathrm{l}\mathrm{o}\mathrm{g}}_3\left(1+\frac{1}{3}\right)+{\mathrm{l}\mathrm{o}\mathrm{g}}_3\left(1+\frac{1}{4}\right)+{\mathrm{l}\mathrm{o}\mathrm{g}}_3\left(1+\frac{1}{5}\right)+......+{\mathrm{l}\mathrm{o}\mathrm{g}}_3\left(1+\frac{1}{242}\right)$ when simplified has the value equal to

If $3^{2\hspace{0.17em}\mathrm{\hspace{0.17em}}{\log }_3\hspace{0.17em}x\hspace{0.17em}}–2x-3=0$ , then the number of values of $x$ satisfying the equation is

The expression ${\left(a^{\alpha }\right)}^{-\beta \mathrm{l}\mathrm{o}{\mathrm{g}}_{a^s}\left({\mathrm{N}}^{\mathrm{\gamma }}\right)}$ where $\left(a > 0\right)$, when simplified reduces to

If $a^2=b^3=c^5=d^6,$ then ${\log }_d\left(abc\right)$ is equal to

If $U=x^3+y^3+z^3-3xyz,$ then $\log U$ is equal to

If $x, y, z$ are $3$ positive numbers not equal to $1$, satisfying $x^2+y^2+z^2-xy-yz-zx=0$ , then

If $xyz = 4,$ then ${\mathrm{l}\mathrm{o}\mathrm{g}}_2\left(\frac{4}{\sqrt{x}}\right)+{\mathrm{l}\mathrm{o}\mathrm{g}}_2\left(\frac{4}{\sqrt{y}}\right)+{\mathrm{l}\mathrm{o}\mathrm{g}}_2\left(\frac{4}{\sqrt{z}}\right)$ is equal to-

$\sqrt{{\left({\log }_{3}4+{\log }_{2}9\right)}^2\hspace{0.17em}-{\left({\log }_{3}4-{\log }_{2}9\right)}^2}$ has the value equal to

If $x, y \& z$ are three consecutive natural numbers, then $\log \left(1+xz\right)$ is equal to