Assuming your IDs are auto numbered, you'd do like this:
SELECT pe.person_id,
pe.name,
al.activity_name,
al.activity_date
FROM person pe
LEFT JOIN (SELECT p.person_id,
Max(a.activity_id) activity_id
FROM person p
LEFT JOIN activity_log a
ON ( p.person_id = a.person_id )
GROUP BY p.person_id) AS LAST
ON pe.person_id = LAST.person_id
LEFT JOIN activity_log al
ON LAST.activity_id = al.activity_id
However, users might enter past activities later than newer ones, then this will fail and you'd have to go like this:
SELECT LAST.person_id,
LAST.name,
LAST.activity_date,
(SELECT activity_name
FROM activity_log al
WHERE al.person_id = LAST.person_id
AND al.activity_date = LAST.activity_date) activity_name
FROM (SELECT p.person_id,
Max(p.name) AS name,
Max(a.activity_date) activity_date
FROM person p
LEFT JOIN activity_log a
ON ( p.person_id = a.person_id )
GROUP BY p.person_id) AS LAST
But this still has a problem: since MySQL does not allow LIMIT in sub-queries, the query will fail if the same person has two activities with the same activity_date and that's the latest date.