log(n!) 和 log(n) 之间哪个增长率更高？

Question

I have just started learning and practicing algorithms and I read in a book that log(n!) has a faster growth rate than log(n).

log(n!) = log((n)*(n-1)*.....(1))  
log(n!) = log(n)+log(n-1)+....+log(1)  <--- (i)

taking the worst case from (i), which is log(n)
therefore, log(n) should have same growth rate as log(n!)

Then how is it that log(n!) is better than log(n)?

Answer 1

Lets try a different approach

1.

n! = n*(n-1)*(n-2)*...*1 n! = n*(n-1)*(n-2)*...*1 < n*n*n.....*n

n! < n^n
log(n!) < log(n^n)
log(n!) < nlogn
** log(n!) = O(nlogn)

---

2.

n! = 
n*(n-1)*(n-2)*....*n/2*(n/2-1)*...*1 > (n/2)*(n/2)*(n/2)*...*(n/2)
                                       ---------------------------
                                                  n/2 times

This is true because:

n > n/2
n-1 > n/2
n-2 > n/2
.
.
n/2 == n/2

So now we know: log(n!) > log((n/2)^(n/2)

log(n!) > n/2*log(n/2)
*** log(n!) = Ω(nlogn)

From ** and *** we get:
log(n!) = θ(nlogn)

Hence log(n) = O(log(n!))