LD_DEBUG=libs ./program 2> log

This produces stuff like this:

1194805:  find library=libasan.so.8 [0]; searching
1194805:   search cache=/etc/ld.so.cache
1194805:    trying file=/usr/lib/libasan.so.8
1194805:  
1194805:  find library=libm.so.6 [0]; searching
1194805:   search cache=/etc/ld.so.cache
1194805:    trying file=/usr/lib/libm.so.6
...
1194805:  calling init: /lib64/ld-linux-x86-64.so.2
1194805:  
1194805:  calling init: /usr/lib/libc.so.6
...
1194805:  calling fini: /lib64/ld-linux-x86-64.so.2 [0]
1194805:  
1194805:  calling fini: /usr/lib/libicudata.so.76 [0]