If your firmware is >2.80, you need a copy of GTA:LCS unpatched (original version). If you have that game, there is a downgrader that works with it and firmware 3.03. You can find a link for it on Dark_AleX's web page
here.
Note: if you follow the guide in this thread for downgrading a TA-082, you might have noticed that the USB connection doesn't show as a PSP anymore, and USBHOSTFS doesn't work anymore. This is because the 2.71 IdStorageChanger app changed more keys than were necessary to get 1.50 to run. You can "fix" these keys with an application I did based on original code from Stapol and others, and info from the lan.sh forums.
First, you need a key 0x0043 from a non-downgraded TA-082. You can use the kxploited mapIDv3 found
here to dump the keys on a non-downgraded PSP (using something like HEND).
Once you have the key, make a directory in the GAME150 directory of the downgraded PSP, and place the EBOOT.PBP from the idreset-v7.zip arc included here in it. Copy the 0x0043.bin file from the mapIDS directory to this directory and rename it 043.bin.
Then run idreset. It will tell what it thinks the status of the PSP keys are, and if they need fixing, it will ask you to press X to fix them. It allows you the option of exiting without changing the PSP idstorage keys, so you don't have to worry on that account. It does nothing without asking first.
You'll find a long thread on the matter
here. I suggest people read the thread before attempting to fix their keys.