Logarithm — NDA Mathematics PYQs

If $1-{\log }_{10}2={\log }_{10}(5^x+4^x+3^x+2^x+1)$, then what is a value of $x$ ?

If ${\log }_{b}a=p$, ${\log }_{d}c=2p$ and ${\log }_{f}e=3p$, then what is $(ace{)}^{\frac{1}{p}}$ equal to?

What is the number of solutions of ${\log }_4(x-1)={\log }_2(x-3)$?

For $x\ge y>1$,
let ${\log }_x\left(\frac{x}{y}\right)+{\log }_y\left(\frac{y}{x}\right)=k$,
then the value of $k$ can never be equal to

The value of $x$, satisfying the equation ${\log }_{\cos x}\sin x=1$, where $0<x<\frac{\pi }{2}$, is

If $n=100!$, then what is the value of the following?
$$\frac{1}{{\log }_{2}n}+\frac{1}{{\log }_{3}n}+\frac{1}{{\log }_{4}n}+\cdots +\frac{1}{{\log }_{100}n}$$

If ${\log }_{10}2{\log }_{2}10+{\log }_{10}(10^x)=2$, then what is the value of $x$?

What is the value of $\frac{1}{10}{\log }_{5}1024-{\log }_{5}10+\frac{1}{5}{\log }_{5}3125$?

If $x={\log }_c(ab)$, $y={\log }_a(bc)$, $z={\log }_b(ca)$, then which of the following is correct?

If $x^{{\log }_{7}x}>7$ where $x>0$, then which one of the following is correct?