His speed is equivalent to that of a horse. OK. Here, 'equivalent' is an adjective.
The dog's age is equivalent of 150 in human age. is this correct?? No. If you want to say something like this, say The dog's age is the equivalent of 150 in human years.Here, you are using 'equivalent' as a noun, so you need an article.
Many questions filled my head, for their expressions were as if there had been a divorce, death, or illness in the family. It was worse. They hit me with what was the equivalent of all three.