From the library website:
Early libraries in Logan were located in two institutions of higher learning: Brigham Young College (founded in 1877) and Utah Agricultural College (founded in 1888 and now Utah State University). Efforts for a public library began soon after the turn of the 20th Century. St. John's Episcopal Church opened a reading room for their parishioners in 1906, and the Cache Stake of The Church of Jesus Christ of Latter-day Saints also had a reading room as early as 1912.
It was understood, however, that these efforts were insufficient. On the 18th of April, 1916, George Thomas, W.R. Owen, and H.E. Hatch appeared before the Logan City Board of Commissioners with a petition to "establish and maintain a public library and reading room", which was approved. 
The current library building had been constructed in 1927 as the Palais D'Or Dance Hall. Its most recent use, before being occupied by the library (and, at the time, the city hall), had been as a Sears Department Store. 
Logan Library. The view is from the back garden area, not the street side.