Given a fixed real number $c$, define a random sequence $(X_n{)}_{n\ge 0}$ by the following random process:

• $X_0=c$ (with probability 1).
• For $n>0$, $X_n=U_n{X}_{n-1}$ where $U_n$ is a real number chosen at random between zero and one, uniformly, and independently of all previous choices $(U_m{)}_{m<n}$.

If we desire there to be precisely a 25% probability that $X_{100}<1$, then this can be arranged by fixing $c$ such that ${\log }_{10}c\approx 46.27$.

Suppose now that $c$ is set to a different value, so that there is precisely a 25% probability that $X_{10000000}<1$.

Find ${\log }_{10}c$ and give your answer rounded to two places after the decimal point.