Embibe Experts Solutions for Chapter: Logarithm and Its Applications, Exercise 2: EXERCISE-2

Which of the following are correct?

If  $p, q\in N$  satisfy the equation $x^{\sqrt{x}}={\left(\sqrt{x}\right)}^x$ then $p$ & $q$ are -

The expression, ${\log }_p {\log }_p \underset{\begin{array}{ccc}n& radical& sign\end{array}}{\underbrace{\sqrt[p]{\sqrt[p]{\sqrt[p]{......\sqrt[p]{p}}}}}}$  where $p\ge 2, p\in N$, when simplified is -

Which of the following numbers are positive?

If  ${\log }_{p}q+{\log }_{q}r+{\log }_{r}p$ vanishes where $p, q$ and $r$ are positive reals different than unity then the value of ${\left({\log }_{p}q\right)}^3+{\left({\log }_{q}r\right)}^3+{\left({\log }_{r}p\right)}^3$ is -

The solution set of the system of equations, ${\log }_{12}x\left(\frac{1}{{\log }_{x}2}+{\log }_{2}y\right)={\log }_{2}x$ and ${\log }_{2}x·\left({\log }_3(x+y)\right)=3{\log }_{3}x$ is -

If $x_1$ and $x_2$ are the solution of the equation $x^{3{{\log }^3}_{10}x-\frac{2}{3}{\log }_{10}x}=100 \sqrt[3]{10}$ then -

If  $a^x=b, b^y=c, c^z=a$ and $x={\log }_b{a}^2; y={\log }_c{b}^3$  and $z={\log }_a{c}^k$ , where $a, b, c>0$ and $a, b, c\ne 1$ then $k$ is equal to -