Well, here's a little bit of background information from historylink.org:

Yesler's Mill, the first steam-powered sawmill on Puget Sound, is under construction in Seattle on October 30, 1852.

HistoryLink.org Essay 5394



On October 30, 1852, the Olympia newspaper The Columbian reports that "a new steam mill is in process of erection by Mr. H. L. Yesler at Seattle." The region's first steam-powered saw mill begins operation in the following spring and quickly establishes Seattle as the economic capital of Puget Sound.

The Columbian was Puget Sound's only newspaper and it was in this issue that the name Seattle first appeared in print. The paper reported:

"Huzza for Seattle! It would be folly to suppose that the mill will not prove as good as a gold mine to Mr. Yesler, besides tending greatly to improve the fine town-site of Seattle, and the fertile country around it, by attracting thither the farmer, the laborer and the capitalist. On with improvements! We hope to hear of scores of others ere long"(quoted in Finger).

Henry Yesler (1810-1892) surveyed Puget Sound during the summer of 1852 looking for the best site for the area's first mill. In early October 1852, Carson Boren (1824-1912) and Dr. David S. Maynard (1808-1873) persuaded him to choose Seattle by donating a waterfront lot called "the Sag" and a corridor down which logs could be dragged from upland forests. This strip was originally named Mill Street but most called it Skid Road; today it is Yesler Way.

In 1878, Yesler told an interviewer working for Hubert Howe Bancroft, the early historian of the West:

My mill was the first steam saw mill put up on the Sound. Lumber sold for $35. a thousand then; now for $10. As there was no wharf the lumber had to be rafted from the mill to the vessels. In unloading the machinery for the mill we had to throw it all into the water & let it float ashore. The boiler was floated in this way, but the engine was placed on a raft. After the establishment of this mill which was commenced in '52, the town grew rapidly. We commenced sawing wood under a shed in March '53 – the saw dust we filled swamps with, and the slabs we built a wharf with. The wharf where you now land is all filled in with slabs" (Yesler narrative).

Sources:
Clarence Bagley, History of Seattle (Chicago: S. J. Clarke Publishing Co., 1916); Bagley, History of King County, Washington (Chicago: S. J. Clarke Publishing Co., 1929); Murray Morgan, Skid Road, An Informal Portrait of Seattle (New York: Viking Press, 1951); John Robert Finger "Henry L. Yesler's Seattle Years: 1852-1892" University of Washington Ph.D Dissertation, 1968, p. 16;

Henry Yesler narrative in "Eldridge Morse Notes on the History and Resources of Washington Territory Furnished to H. H. Bancroft (ca 1880)" [P-B 30-54], Hubert Howe Bancroft Collection, Bancroft Library, University of California Berkeley, Berkeley, California: Microfilm Edition, Newspapers and Microforms, Reel No. ? , University of Washington Libraries, Seattle, Washington.

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I was hoping to find this in Linda Allen's Washington Songs and Lore songbook, but 'tain't there.