Mfm9's suggestion for logging the index is a very good way to minimize memory usage. But if you need to log all invalid entries for security reasons then you need all the digits (and probably lots of RAM to hold them).
There are lots of tricks you can do to reduce the number of bytes to hold time, for example if a minute is sufficient precision then you could use two bytes to hold the number of minutes since the last logged reading (you would store the full four digit date and time of the first reading only). But it would be prudent to keep things simple to start off with.