After eight years of waiting, George Bush's patience, loyalty, and hard work finally paid off for him this week with the Republican presidential nomination.

But, now that his political apprenticeship to President Reagan is over, Bush faces a set of challenges that in some ways are much tougher. Tougher because, if they are to be surmounted, the job must be done in only a few weeks. Tougher because the challenges involve some problems and misperceptions that have been many years in the making.One challenge is to overcome Bush's well-known but badly overblown image for supposedly not being sufficiently firm and decisive. Bush must convince Americans that he can be not just a loyal vice president but an inspiring leader.

Another challenge is to hold on to the Democratic voters whom President Reagan won over, a task that GOP strategists concede won't be easy without Mr. Reagan's unique brand of personal political charm. Part of the problem is that with more defense spending and little inflation, the Republican ticket no longer has some of the main issues that helped bring the Reagan-Bush team to victory.

Still another challenge is for Dan Quayle to demonstrate that, at 41 years of age and after two terms in the House and two in the Senate, he has what it takes to be a heartbeat away from the presidency.

Mastering these challenges will take more than just a couple of speeches winding up the GOP National Convention in New Orleans, no matter how vigorous and impressive those performances may have been.

Newsmen who have followed Bush's long career rate Thursday night's acceptance speech the best he has ever given. Speaking slowly but forcefully, Bush projected strength and confidence, displayed good humor, and outlined his vision of an America building on the accomplishments of the Reagan administration. In short, he accomplished his objective of looking presidential.

Though Quayle also did well with his address, the GOP convention ended with a shadow hanging over his selection because of disclosures that he used his family's influence to get into the National Guard instead of going to Vietnam during that conflict. As "scandals" go, this episode from 19 years ago may be damaging but certainly does not seem fatal to the Bush-Quayle campaign.

Anyway, can the various other challenges to the ticket be surmounted?

Part of the answer was provided during the convention when Bush's image was considerably improved not just by his solid acceptance speech but by the way he was humanized through the strength and love he so obviously draws from his family.

Another part of the answer is provided by the public opinion polls. While Democratic nominee Michael Dukakis holds a lead in the polls, it keeps going up and down. The latest polls show the Republican convention gave new impetus to Bush's candidacy, with the result that the presidential race is now virtually a dead heat. This fluctuation indicates that much of the support for each of the tickets is weak at this point and that plenty of voters can be swayed.

Still another part of the answer is provided by common sense. It's insulting to women to think that the gender gap facing Bush - which is to say the poll results showing most women prefer Dukakis - could be overcome just by adding Quayle's handsome face to the GOP ticket. And it's insulting to Bush to think that he doesn't grasp such an obvious point.

Moreover, the presidential election campaign is about much more than competence - Bush's experience as congressman, U.N. ambassador, CIA chief, and vice president versus Dukakis' experience as governor of Massachusetts. It's also about ideas, programs, and policies.

On this score, there are some significant differences between the parties and candidates. That much should be clear not just from the campaign speeches given so far, but also from the GOP and Democratic platforms. The Democrats' is short and vague, lacking the laundry list of promises that backfired on them in 1980 and 1984. By contrast, the Republicans' is six times longer and quite specific, addressing by one count 125 issues that the other platform does not mention.

Among the differences: The Republicans condemn abortion, favor school prayer, oppose the Equal Rights Amendment, and frown on the concept of comparable worth under which the government would determine how much various jobs should be paid. The Democrats take the oppose stance on these issues. On national security, the GOP would promptly deploy the MX missile and a space-based missile defense, while the Democrats would scrap these programs. On taxes, the GOP platform rules out a boost in income levies and calls for lowering the capital gains tax; the Democratic platform is silent on taxes.

With such differences also comes the potential for a campaign that could be more than just rowdy. Now that the conventions are over, the main challenge facing both tickets is to educate the voters with a campaign that is clean but still vigorous and candid.