Suppose that

${\log }_2\left({\log }_3\left({\log }_5\left({\log }_{7}N\right)\right)\right)=11$

How many different prime numbers are factors of $N$?

Suppose that

${\log }_2\left({\log }_2\left({\log }_{2}x\right)\right)=2$

How many digits are in the base-$10$ representation for $x$?

Let $f\left(n\right)={\log }_{2002}{n}^2$ for all positive integers $n$. Define $N=f\left(11\right)+f\left(13\right)+f\left(14\right)$

Which of the following relations is true?

For some real numbers $a$ and $b,$ the equation

$8x^3+4a{x}^2+2bx+a=0$

has three distinct positive roots, and the sum of the base-$2$ logarithms of the roots is $5.$ What is the value of $a$?

For any positive integer $n,$ define

$f\left(n\right)=\left\{\begin{array}{ll}{\log }_{8}n,& \text{if} {\log }_{8}n \text{is rational}\\ 0,& \text{otherwise}\end{array}\right.$

What is $\sum _{n=1}^{1997}f\left(n\right)$?

What is the value of the expression

$N=\frac{1}{{\log }_{2}100!}+\frac{1}{{\log }_{3}100!}+\frac{1}{{\log }_{4}100!}+\dots +\frac{1}{{\log }_{100}100!}$?

What is the value of the sum

$S={\log }_{10}\left(\tan 1°\right)+{\log }_{10}\left(\tan 2°\right)+\dots +{\log }_{10}\left(\tan 88°\right)+{\log }_{10}\left(\tan 89°\right)$?

Let $a\ge b>1.$ What is the largest possible value of

${\log }_a\left(\frac{a}{b}\right)+{\log }_b\left(\frac{b}{a}\right)$?