You say "You get what you pay for."
Sure, you could get a logo for your business for $5. You could also build your website with the free "website builder tool" which your registrar offers, and you could get your business cards designed with the "add an element and push print" app from the mass printing company.
The question is this: will these el cheapo products be any good, or will they look el cheapo?
People know about Fiverr, the website builders, and the mass printers. People generally know the quality of the end product, which is somewhere from crap to mediocre.
So you tell your client "The difference is that companies using these services don't look professional. They look, frankly, like startups working out of the basement. If you're just going for local customers who don't care that you're a small operation with three other customers, go for it, and best of luck. If you want to build a business which is going to grow and attract real customers and real volume, you need to invest in a professional logo and a quality corporate identity."
Consider this too: from a work standpoint, do you want the customer who's going to object to paying an appropriate fee for an appropriate amount of work, and constantly nickel-and-dime you and bitch about costs? Or do you want to court the companies which understand that advertising is an investment and not an expense?
And yes, I have walked away from clients with this attitude. It's not worth the time or acida.
ETA something else I thought of: Don't be a jerk to the client when making this statement. The client may be simply uneducated rather than a bean-counter. Leave the door open: "If this [the $10 logo] works for you now as a small business, great. In a year or two, if you decide you want to upgrade, I'd love to hear from you and help you expand/improve/rebrand your business." Present it as a business decision, with pros and cons, and if this is just someone who's starting out rather than someone who will nickel-and-dime everyone over everything, the client might be willing to come back when they're ready to invest in a more professional look.