When you buy a home in California, almost always the purchase price is the basis for your assessed value. If you buy as your owner occupied residence, then you can claim a homeowners exemption for $7,000 here in CA, meaning if you buy a home for $300k and file that exemption then the assessed value would really be $293k.
The treasurer then uses that assessed value to tax the property. California has a basic levy tax rate of 1% ($2,930/year based on a $293k assessed value), and then on top of that there may be municipal bonds, school district & water district taxes, etc (usually these are flat dollar amounts, not based on a % of the assessed value).
The new tax rate is effective that day you buy the home, but it doesn't change right away, what happens is that when the assessor gets notified of the new purchase price they will update their assessment records, the treasurer will recalculate the property taxes based on the new assessed value, and you will be sent out a supplement tax bill (credit) for the amount you underpaid/overpaid in the time since you purchased the home up until your supplement tax bill/credit was sent.
is where you can find LA County property tax rates (as well as a link to look up the parcel number for a specific property you want to look up the property tax rate for - it's a 2 step process).