Some years ago, I applied to a SE asian airline as a B707 Captain.
I was promptly accepted and the DGCA issued the local relevant ATPL license
because I had, on application, a stamped/approved pilots log book with the relevant command hours duly recorded.
Another applicant, who did not have any log book, presented to the DGCA computer copies of his previouis flying hours at JAL.
Not signed log book, no license.
Opps!
Better safe than sorry.
IE: keep youir personal log book right up to date and have it stamped/signed by the company, for best results.
Who knows...one might find greener pastures elsewhere and need the log.