Let a(1),a(2),a(3), …, a(10) be in G.P. ...

Let `a_(1),a_(2),a_(3), …, a_(10)` be in G.P. with `a_(i) gt 0` for i=1, 2, …, 10 and S be te set of pairs (r, k), r, k `in` N (the set of natural numbers)
for which `|(log_(e)a_(1)^(r)a_(2)^(k),log_(e)a_(2)^(r)a_(3)^(k),log_(e)a_(3)^(r)a_(4)^(k)),(log_(e)a_(4)^(r)a_(5)^(k),log_(e)a_(5)^(r)a_(6)^(k),log_(e)a_(6)^(r)a_(7)^(k)),(log_(e)a_(7)^(r)a_(8)^(k),log_(e)a_(8)^(r)a_(9)^(k),log_(e)a_(9)^(r)a_(10)^(k))|` = 0. Then the number of elements in S is

Infinitely many

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