Trying to optimize a MySQL query that has sub selects
Try runnning the LIMIT 50 against the client table earlier
SELECT c.identity, c.firstName, c.lastName, c.email_address, c.startDate,
(SELECT COUNT(identity) FROM ClientLog cll WHERE cll.clientID = c.identity AND cll.type = 2) as loginCount, 
(SELECT COUNT(identity) FROM ClientLog clv WHERE clv.clientID = c.identity AND clv.type = 10) as viewCount
FROM (SELECT * FROM client LIMIT 50) c;

or gather the client keys and join a little differently
SELECT c.identity, c.firstName, c.lastName, c.email_address, c.startDate,
(SELECT COUNT(identity) FROM ClientLog cll WHERE cll.clientID = clientkeys.identity AND cll.type = 2) as loginCount, 
(SELECT COUNT(identity) FROM ClientLog clv WHERE clv.clientID = clientkeys.identity AND clv.type = 10) as viewCount
FROM
(SELECT identity FROM client LIMIT 50) clientkeys INNER JOIN client c USING (identity);

You have to control the subselects by giving them as little data as possible. That's why I suggested running LIMIT 50 before joining to the correlated subqueries.
UPDATE 2012-06-15 12:18 EDT
Since this is a full join, remove the LIMIT 50 from your original query:
SELECT c.identity, c.firstName, c.lastName, c.email_address, c.startDate,
(SELECT COUNT(identity) FROM ClientLog cll WHERE cll.clientID = c.identity AND cll.type = 2) as loginCount, 
(SELECT COUNT(identity) FROM ClientLog clv WHERE clv.clientID = c.identity AND clv.type = 10) as viewCount
FROM (SELECT * FROM client) c;

and make sure ClientLog has a decent index for the correlated subqueries
ALTER TABLE ClientLog ADD INDEX clientID_type_index (clientID,type);

Give it a Try !!!