@Ian M. and @Ryan Zomorodi You are permitted to file a tax appeal between Jan 1st and Mar 31st every year (you have 2 more months). This is handled by the Cuyahoga Board of Revisions which is part of the Cuyahoga County Treasurer's office. You can do this personally but you will need to appear at a hearing so most of my clients use an attorney to file it for them. I have two different attorneys that do a lot of these, PM for their details.
The value is always considered as of the "tax lien date" which is January 1st of whatever year you are protesting. If you can establish that it was an Arms Length Transaction between two independent parties they should use the purchase price.
I say if and should because the board tries to take a stance that an REO transaction is NOT arms length. While I can understand that a sheriff sale could be deemed as a forced transaction (depite the fact that people are openly bidding), there seems to be no rhyme or reason why a REO sold from a (sophisticated) bank using the MLS for marketing is not an arms length transaction. However, be prepared that you may not get all the way down to your purchase price on an REO but you are likely to get some type of reduction.
In your example of $120,000 current tax value and $70,000 purchase, they might come back with an $80,000 or $90,000 value as a compromise. There are several boards and they also act a bit differently.