A radioactive substance decays at a rate proportional to the amount of substance present. If half of the substance decays in 100 years, then what is the decay constant (proportionality constant) ?

If log10 2 log₂ 10 + log₁₀(10^x) = 2, then what is the value of x?

The value of x, satisfying the equation  $log_{cos x} ~sin x = 1$, where $0<x<\dfrac{\pi}{2}$, is

If n = 100!, then what is the value of the following?

$\rm \dfrac{1}{log_2n}+\dfrac{1}{log_3n}+\dfrac{1}{log_4n}+{.....}+\dfrac{1}{log_{100}n}$