The highly competitive web-search business became a lot more interesting today when Microsoft Corp. made an unsolicited bid to purchase Yahoo Inc. for $44.6 billion–far more than Yahoo shares were worth when the stock market closed yesterday.

And as Microsoft’s dramatic move was clearly aimed at strengthening its competitive edge against Google Inc., the prospect was for much greater competition between the two online powerhouses.

That, in turn, seemed destined to improve the internet experience for school officials, ed-tech leaders, teachers, and students among the many millions of internet users around the world for whom web searching and related software applications have assumed increasing day-to-day importance.

Microsoft’s unexpected announcement came as both it and Yahoo were struggling against frontrunning Google in the race to capture online advertising dollars. Analysts also saw the prospective deal as giving a positive jolt to the entire technology market.

In a letter to Yahoo’s board of directors, Microsoft CEO Steve Ballmer said his company would bid $31 per share, representing a 62 percent premium to Yahoo’s closing stock price Thursday. The offer sent Yahoo’s share price up 54 percent in premarket trading, while Google fell 8 percent.

Since reaching a 52-week high of $34.08 in October, Yahoo shares have fallen 46 percent. Yahoo rose $10.27 to $29.45 in premarket trading today, and by midday it stood just below $28 a share. Google, meanwhile, was down about 9 percent and Microsoft was off nearly 8 percent.

Ballmer said in his letter that Yahoo had told the world’s biggest software company a year ago that the Yahoo board felt it was not the right time to enter into discussions regarding a deal.

The main reason for that view was attributed to the Yahoo board’s confidence in realizing “potential upside” based on a new operational strategy and “a significant organizational realignment.” Ballmer added: “A year has gone by, and the competitive situation has not improved.”

According to the proposed deal, Yahoo shareholders could choose to receive cash or Microsoft common shares, with the total purchase consisting of 50 percent each in cash and stock.

Microsoft said the merger would generate at least $1 billion in savings. The company planned to offer significant retention packages to Yahoo engineers, leaders, and employees. The company said it believes the takeover would receive regulatory approval and close in the second half of this year.

For the educators and the rest of the internet world, the prospective combination of Microsoft and Yahoo against Google represents the latest major development in a high-stakes “battle for the desktop.” With personal, professional, and educational computing becoming increasingly sophisticated, big-name technology companies have been fighting fiercely for marketing supremacy–leading to big gains for so-called software as a service and other internet-based products. See Web offerings spread in ‘battle for desktop.’

As Microsoft made its offer for Yahoo, Ballmer said his company expected Yahoo’s board will review its proposal, but “reserves the right to pursue all necessary steps to ensure that Yahoo’s shareholders are provided with the opportunity to realize the value inherent in our proposal.”

Microsoft’s announcement followed one by Yahoo late Thursday that Terry Semel was stepping down as its chairman, severing his ties with Yahoo 7 1/2 months after resigning as its chief executive under shareholder pressure. Semel had been criticized for failing to cash in sufficiently on the web advertising surge in comparison with Google.

Yahoo co-founder and Chief Executive Jerry Yang said this week the company will cut 1,000 jobs, or 7 percent of its work force, in an effort to cut costs. Meanwhile, Microsoft last week forecast a rosy 2008 despite broader economic worries after it blew past Wall Street’s expectations for a second consecutive quarter.