Logarithms Questions (7)

Least positive integral value of $a$ for which $\log_{(x+1)/x}(a^2-3a+3)>0$ for all $x>0$

$\displaystyle\lim_{n \to \infty} \dfrac{n^2}{\left((n^2+1^2)(n^2+2^2)\cdots(n^2+n^2)\right)^{\frac{1}{n}}}$ equals:

Least positive integral value of $a$ for which $\log_{(x+1)/x}(a^2-3a+3)>0$ for all $x>0$

Let $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ satisfy the system: $\log_{10}(2xy)=4+(\log_{10}x-1)(\log_{10}y-2)$ $\log_{10}(2yz)=4+(\log_{10}y-2)(\log_{10}z-1)$ $\log_{10}(zx)=2+(\log_{10}z-1)(\log_{10}x-1)$ with $x_1>x_2$. Match List-I with List-II: P) $y_1/x_1$; Q) $z_1\cdot x_2$; R) $z_1\cdot x_2/z_2$; S) $y_2+z_1\cdot x_2$

Let $(x_1,y_1,z_1)$ and $(x_2,y_2,z_2)$ satisfy the system: $\log_{10}(2xy)=4+(\log_{10}x-1)(\log_{10}y-2)$ $\log_{10}(2yz)=4+(\log_{10}y-2)(\log_{10}z-1)$ $\log_{10}(zx)=2+(\log_{10}z-1)(\log_{10}x-1)$ with $x_1>x_2$. Match List-I with List-II: P) $y_1/x_1$; Q) $z_1\cdot x_2$; R) $z_1\cdot x_2/z_2$; S) $y_2+z_1\cdot x_2$

Least positive integral value of $a$ for which $\log_{(x+1)/x}(a^2-3a+3)>0$ for all $x>0$

If $\log_{12}27=a$, then $\log_6 16=$