First of all, it depends on what you mean by "profit."
Your 50% share of the profit is $30,000, but if that's the total of your income from the salon and you're working 40 hours to make it, then your business is not really earning a "profit" at all. It's simply generating a wage or salary that you could make working somewhere else without the risks of ownership.
It's only a profit it the number takes into account the value of your time in running and working at the business.
Assuming that your salon truly generates a $60,000 economic profit, though, your valuation is likely to be based on some multiple of this number.
For a really quick (and barely reliable) way to do this, you can go to a place like BizBuysell-dot-com and look up the other salons that are for sale. There will be a listing price, the company's revenue, and the company's cash flow. You can then try to see if you can determine an average multiplier like "1x cash flow" or "2x revenue" or something like that.
This is by no means 100% accurate as (1) people are usually not getting valuations done when they sell their businesses online, and they are overly optmistic about their business' value, (2) cash flow can be defined lots of different ways, so you don't know if this will be a true "apples to apples," and (3) sometimes business include real estate or other fixed assets that can jack up the price, making them look more valuable on a multiple of cash flow.
This business is probably not valuable enough to do a full-tilt valuation on, but I bet if you look around you can find a good multiplier of cash flow that you and and your partner can agree on.
Another approach would be to ask yourself -- and answer honestly -- if your partner were selling the 50% share to you instead of the other way around, what would you be willing to pay? If you are being honest and fair, you should expect to ask for no more from your partner for your own 50% share.