SOLUTION: http://prntscr.com/d1qng (See the screenshot)

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Question 634065: http://prntscr.com/d1qng (See the screenshot)
Answer by jsmallt9(3758) About Me  (Show Source):
You can put this solution on YOUR website!
log%28x%2F9%2C+%28x%5E2%2F3%29%29%2C+6+%2B+log%283%2C+%289%2Fx%29%29
I'm not sure how this problem could be done algebraically. If it is possible, I imagine that knowing log%28x%2F9%2C+%289%2Fx%29%29+=+-1 might be helpful.

Below I have gotten algebra.com's graphing software to graph
y+=+log%28x%2F9%2C+%28x%5E2%2F3%29%29 (in red) and
y+=+6+%2B+log%283%2C+%289%2Fx%29%29 (in green)
As you can see, the red graph is below (i.e. less than) for x's between 0 and 9. So there are 8 positive integers, 1 through 8, that would fit the inequality.