Generally, it's presumed students know about directed numbers (which may or may not be considered algebra), because otherwise they can't use logarithms of numbers less than 1.
But apart from that proviso, the answer is that this was absolutely the case in some countries, before calculators replaced logarithm tables. For students aiming for O-levels in England (taken at age 16 by roughly the top 25% in overall academic ability), logarithms were typically introduced at age 12-14, with an emphasis on practical calculations but little on theory. While it is also true that students were acquainted with some algebra at this age, logarithms were generally presented only numerically at this stage.
For example, if you look at the old O-level syllabi of one examining board, you'll see that logarithm tables were used in exams as late as 1984, and they were explicitly part of the arithmetic syllabus as late as 1974.
For examples of textbooks with early introduction of logarithms, you could have a look at General Arithmetic for Schools by Durell, or Arithmetic, Part II by Siddons, Snell and Lockwood.