If binary_search(array, i, n, value)
searches elements i ... n of the array for value
using binary search, then yes, your analysis is correct. The runtime will be
Θ(log 1 + log 2 + log 3 + ... + log n) = Θ(log n!)
Note that by Stirling's approximation, log n! = Θ(n log n), so the total runtime would be Θ(n log n).
Hope this helps!