I don't think it's quite right to say logic equates to the truth, but it is the means by which we verify truth. It is also how we formulate ideas to begin with, even the construction of a linguistic sentence is more a less a logical process.
But do look at this way that may favor the OP. Does truth exist in any inherent sense? Is truth a part of reality, or strictly a subjective human concept? If it's strictly subjective, than perhaps logic, by being the means of determining truth, is in fact truth.