The appendix A in the Common Core State Standards for Mathematics seems to agree that traditional curricula put logarithms somewhere after Algebra I. CCSSM isn't a curriculum, and it doesn't suggest teaching logarithms later, it just offers "pathways" that are supposed to help districts using traditional curriculum find a way to meet the standards. I mention it just to show that, yes, this seems to agree that a traditional time to introduce the subject is later than Algebra I.
The Massachussetts Curriculum Frameworks put logarithms into "Algebra I" under a category like "Functions." Although this is also considered to be "Common Core Aligned."
One short answer is that "it's traditional." The order you describe is seen as early as 1898 in Bull Wentworth's New School Algebra. (Donoghue, 2003)
Curriculum design is (should be) about structuring the subject matter in a way that students can form a coherent understanding. At the least, it is an arrangement that the designers find coherent. This means that it's not necessarily true that the options are: square roots are more common, square roots are more useful, or square roots are easier. Nor is it necessarily true that there is one correct way. So, one way may be produced, become traditional, and then be perpetuated.
Reform curricula have come up against this time and again. Sometimes, the way we do things in math education are traditional and people don't want to question them, even in the face of evidence.
While I think you have a good question, perhaps a more forward-looking question might be whether we can find reasons to teach logarithms earlier.
Squares and square roots are connected to a particular 2D representation (squares) that can be used as a model for multiplication. It's popular for assisting students beyond the tenacious idea of multiplication as repeated addition. A representational leg up like that may be (just speculating here) one convenient reason that squares and square roots can be placed early in a curriculum "pathway." I'm really only guessing here; IANACD (I Am Not A Curriculum Designer). But I doubt that's a hard and fast rule that cannot be broken. Some people have successfully introduced algebraic thinking to early grades students (Smith III & Thompson, 2008). Although getting students to reason algebraically is a little bit different (in intention) from introducing a particular concept or operation at a different time.
Donoghue, E. F. (2003). Algebra and geometry textbooks in twentieth-century America. In G. M. A. Stanic & J. Kilpatrick (Eds.), A history of school mathematics (Vol. 1, pp. 329–398). Reston, VA: National Council of Teachers of Mathematics.
Smith III, J. P., & Thompson, P. W. (2008). Quantitative reasoning and the development of algebraic reasoning. In J. Kaput, D. W. Carraher, & M. L. Blanton (Eds.), Algebra in the early grades (pp. 95–131). New York, NY: Taylor & Francis Group.