Note: This problem is not suitable for hints!!
This problem was quite fun to solve, and I am lucky I took a statistics class last year.
Below are the main 2 facts that you need to solve this problem
With this we can begin to solve the problem
Step 5 is the crucial step, we make use of (1) by recognising that -ln(U_i) ~ Exp(1) and (2) by recognising that is this is just a sum of 10,000,000 exponential variables. Initially I solved this problem using an online tool (https://homepage.divms.uiowa.edu/~mbognar/applets/gamma.html) and then dividing it by ln(10) (because our answer will be in the form ln(c)), but found an python function (https://docs.scipy.org/doc/scipy/reference/reference/generated/scipy.stats.gamma.html#scipy.stats.gamma) which does the job!
Enter a number (yourinput)
Code will output log_10(c) given that P(X_yourinput < 1) = 0.25