I think it will be a good idea if you send an email to the guys who publish the HCMR to inquire about the original source of the problem, which was proposed by Shrenik Shah, as you might know. I do think there might be a solution involving just high school math, though I don’t have such a solution right now with me.

..involving nothing but (high school) algebra is it possible or not given that it involves the log function?

Here is the problem/solution section that carried the solution by Greg Price. (The pdf file is considerably smaller than the one containing the whole issue; so, it should take much less time to download.)

Greg’s solution uses Taylor series expansion of a logarithmic function and relies on differentiation. I am not sure if you will find it elementary enough.

In fact, “the proposer noted that the inequality in the problem was obtained by taking the limit of the AM-GM-HM inequality for an arithmetic sequence of n terms from a to b as n goes to infinity.” And, I believe that approach is an elementary one, but I think it isn’t “strong” enough to establish strict inequality.

can the problem be solved using only basic algebra and no calculus or analysis?