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Question

Show that: 2(loga4ab+logb4abloga4b/a+logb4a/b).logab
={2,ba12logab,1<b<a

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Solution

Since loga4ab+logb4ab
=14loga(ab)+14logb(ab)
=14(1+logab+logba+1=  logab+1logab+24
=   ⎜ ⎜|logab|+1|logab|2⎟ ⎟2
=12(|logab|+1|logab|)
Consider loga4(b/a)+logb4(a/b)
=14loga(ba)+14logb(ab)
=14(logab1+logba1)= logbb+1logab24
=|logab|1|logab|2
Now, P (say) =loga4ab+logb4abloga4b/a+logb4a/b
P=12|logab|+1|logab|+∣ ∣|logab|1|logab|∣ ∣
Case I : If ba1,then
P=12{|logab|+1|logab||logab|+1|logab|}

=1logab
2Plogab=21=2
Case II : If 1 < b < a, then
P=12{|logab|+1|logab|+|logab|1|logab|}
=logab
2Plogab=2logab

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