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Question

Given a geometric progression
a,a1,a2,a3,.........
and an arithmetic progression
b,b1,b2,b3,.........
with positive terms. The common difference of A.P. and common ratio of G.P. are both positive. Show that there always exists a system of logarithms for which
loganbn=logab (for any n)
Find base β of the system

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Solution

Let r be the common ration of G.P. and d the common difference of A.P.
Then an=arn ....(1)
and bn=b+nd .....(2)
Taking logarithms of both sides of (1) to base
β(β1,β>0), we get
logβan=logβa+nlogβr
logβanbn=logba+nlogβrbnd. by (2).....(3)
Now in order that right hand side of (3) reduces to
logβab, we must have
nlogβrnd=0
or logβr=dorr=βd that is ,β=r1/d
Hence there exists a system of logarithms to base r1/d
such that loganbn=lognab

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