As a small business owner, I would strongly disagree.
It's not a question of if jobs are lost in the city, it's how few will remain after the law is fully enacted?
The restaurant and convenience industry, that depend on low wage employees, will see either enact significant price increases or go out of business. Other industries will leave town and new ones will not move into the city to replace them.
For me to cover the cost of a minimum wage ($7.25/hr) employee and still make a profit, I need to charge my clients' ~$65 per hour for this employee's time. At $15 per hour, and all other things being equal, that cost rises ~$76 - about an $11 increase.
While $11 may not result in much of a price increase when you split it over a 100 burgers, packs of gum, widgets or whatever, it rises exponentially when you start talking about a larger group of employees. With 5 employees, that rate would increase by 2/3 and with 10 employees, it would need to nearly double.
So, fast food and convenience aside, any other business within the city is essentially stuck with 2 choices - leave town or go out of business because businesses outside of the city limits will still be competition. However, these businesses won't have to play by the same rules.