How can I fix my invisible call log?

You can try to do soft reset (Just press and hold both the Sleep/Wake and Home buttons simultaneously for at least 10 seconds, until you see the Apple logo. You can let go of both buttons after the Apple logo appears)

If it doesn't work

You can try to reset settings it will not erase any of your data but will erase some of your settings

So its good to back it up before you do reset.

( Goto Settings --> General --> Reset --> Reset All Settings)

and the last option is restore your iPhone

I hope it helps .