Your logic presupposes two assumptions that I do not agree with, and draws a conclusion that ignores the downside. On the whole, $25 minimum wage is unsupportable.
First, someone on minimum wage should not expect to be living in an apartment on the own - driving a car. I'm not sure when it came to pass that minimum wage was equated to a living wage, but Pres. Obama seemed to draw that conclusion with his $10.10 minimum wage - the figure he uses that would provide a worker with the funds for themself and two dependents.
Second, life in Tennessee is different from life in San Francisco. The folks in Knoxville chose to have lower taxes, less social services, etc. to fit their life choices, just as the folks in San Francisco choose to have very high taxes. The people who own the buildings there choose to offer their apartments at high rents because people are desperate enough to pay them. Food is expensive there because the workers want exotic food.
Your conclusion that $25/hour fixes a lot of things is short sighted. Sure, it would make some of the lowest paid workers there happy for awhile, but Yelp and other employers would do one of two things: either reduce their workforce or move it entirely somewhere else. And rents for everyone would go up. As would tolls, gas prices, etc.
For evidence on layoffs, look what's happening in Seattle - as there minimum wage goes up (beyond what the market used to support), unemployment is going up. That's right - while the national and even regional unemployment numbers are approaching record lows, unemployment in Seattle is increasing.
It's all supply and demand. If more workers can now afford the $1600 rent, there will be a shortage of apartments and owners will charge more. They will have to charge more because their burger prices went up so that the restaurant can pay their busboys $25/hour.