Not Veterinary Advice: No. Don't use Ajax as dogs have very sensitive skin. While dish soap or your favorite shampoo might strip away the dirt, and more importantly the odor, from your pet's coat, it will also strip natural oils from their fur and may irritate their skin.
As per Wsu.edu, a mild hypoallergenic soap that's formulated for veterinary use is all you need. “Formulated for veterinary use” means a product that’s designed to work with a dog’s body.
All grooming products (human and animal) are designed to maximize cleaning and minimize irritation. Human products work best on human skin and veterinary products are designed to work best on dog skin. The chemistry of a dog's skin and fur are different than the chemistry of a human's skin and hair.
Just as humans require good grooming habits, so do our dogs. Besides being a healthy habit for our dogs, grooming is an important part of the relationship we have with them. Check out some grooming tips for dogs at Animalhumanesociety.org.