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If your Lexus still has teh original battery, that is the most likely source of your starting problem. Original equipment batteries have a life expectancy of about 5 years. Since yours is now going on 7 years old, it is well past the age at which it should have been replaced.
Of course a corroded battery cable connection is also a possibility, but the mechanic that replaced the starter would likely have checked for that condition before installing a new starter (but then you would think they would have load tested or replaced the battery at that time as well).
If you need to replace the battey, I would suggest having a top quality professional line of battery installed and not the cheapest one you can find. Batteries come in different quality grades. A top quality battery such as Interstate, Motorcraft, or AC-Delco professional series can be expected to last another 5 to 7 years without any problems. A cheaper grade of battery such as everstart or Duralast usually only lasts about 2 to 3 years in our experience.
Also, make sure they check your charging system for proper operation when it is in for repair. If the charging system is not keeping the battery fully charged it can shorten battery life and cause starting problems such as you are experiencing.
If it is cranking slow, it is either a battery, battery cable, or starter. The only other possibility could be an internal engine problem, but if that were the case it would be making a lot of noise and not running well once it did start.
If your battery is original, there is a very good chance that the battery is the source of the problem.