D.C. City Council Chairman Arrington Dixon yesterday became the third major District of Columbia Democratic leader to endorse President Carter for reelection in a city where many expect Sen. Edward M. Kennedy to handily win the Democratic presidential primary.

Speaking at a press conference in his District Building office, Dixon conceded that it would be "a very, very tough fight" for Carter to win the May 6 primary here.

But Dixon said his endorsement was based on logic rather than "emotion" and "the commitment that President Carter has already been demonstrating."

"I can truly see the promise of full home rule for the District being realized when Jimmy Carter is elected to a second term as president of the United States," Dixon said.

Mayor Marion Barry has offered conditional statements of support for Carter and has attended supporters-only affairs given by Carter's fund raisers. One of those fund raisers was coordinated by Carter's other principal supporter in the city, Robert B. Washington Jr., the D.C. Democratic state chairman.

Three of Dixon's Council colleagues -- at-large members Betty Ann Kane and John Ray and Ward 6 representative Nadine P. Winter -- have already endorsed Kennedy, along with nearly half the members of the D.C. Democratic State Committee, the policy-making arm of the city's regular Democratic organization.

Some local political observers believe the public support for Carter is based mostly on the city government's status as a financial stepchild of the federal government.

Dixon's endorsement of Carter was no surprise. State Committee Chairman Washington is one of the council chairman's closet advisers. Dixon had already begun looking for support for Carter's reelection among some of his council colleagues. In addition, Dixon's father-in-law, Superior Court Judge Carlisle E. Pratt, was appointed to the bench by Carter.

Dixon's announcement yesterday leaves Del. Walter E. Fauntroy as the only leading city Democrat not yet on record in support of Carter's reelection.