Computing Value of n

Question

I am working to compute the value of n for the following equations:-

1. n! = 10^7 (For this computation what i've done so far is n(n-1)! = 10^7 . But how should i proceed further. I have no idea)
2. n^n = 10^9 (For this i tried to take logs on both sides. But the answers are not satisfactory)

Help me out in these computations for the value of n. I want my answer should be computed in terms of Logs or Powers etc..

Answer 1

Try all n, starting from 2 upwards. You will find for 1.):

10! = 3628800 < 10^7 < 11! = 39916800   -> 10<n<11

and for 2.):

9^9 = 387420489 < 10^9 < 10^10 -> 9<n<10

The precise result for 2 is

n=9.295

as computed by wolframalpha .