To log memory usage every second:

sar -r 1

To log memory usage every minute:

sar -r 60

Running sar -r 1 will give the following output to "standard output" (which is your terminal when you use the command from the terminal):

[samuel ~]$ sar -r 1
<snip some computer info>

10:54:08 kbmemfree kbmemused %memused kbbuffers kbcached kbcommit %commit kbactive kbinact
10:54:09 1423000 6595688 82,25 542304 1913592 8017084 49,34 4009672 1572904
10:54:10 1423080 6595608 82,25 542304 1913588 8017084 49,34 4009980 1572904
10:54:11 1422580 6596108 82,26 542304 1913588 8017084 49,34 4009984 1572904
10:54:12 1422396 6596292 82,26 542304 1913588 8017084 49,34 4010160 1572912
10:54:13 1422680 6596008 82,26 542328 1913556 8017084 49,34 4010308 1572888
10:54:14 1428440 6590248 82,19 542340 1907840 8011624 49,31 4010468 1567212
10:54:15 1428076 6590612 82,19 542340 1907836 8012644 49,32 4010520 1567180
10:54:16 1428080 6590608 82,19 542340 1907836 8012644 49,32 4010632 1567180

(Unfortunately the window here on coderwall is a bitt too narrow, so the lines are broken. On the terminal it looks much nicer!)

I find myself using especially the 4:th column, %memused, to quickly have a glance at if I'm about to outrun all of my RAM.

If you want to log the output to a file, you can of course either pipe it directly to a file:

sar -r 1 > somefile.log

... or run it through the tee command, so you can see the output at the same time:

sar -r 1 | tee somefile.log

The sar (System Activity Report) tool is available on Ubuntu via the sysstat package (So, to install on Ubuntu, do: sudo apt-get install sysstat).