Prove by PMI

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Please find below the solution to the asked query:

We have Pn:logxn=nlogxStep 1: Base CaseFor n=1logx1=logxlogx=logx which is true.Hence Pn is true for n=1.Step 2: Inductive hypothesisLet Pn be true for n=k logxk=klogx ;iStep 3: Inductive CaseConsiderlogxk+1=logxk.x=logxk+logx  As logab=loga+logb=klogx+logx  Using i=k+1logxlogxk+1=k+1logxHence Pn is true for n=k+1Since Pn is true for n=1 and assumption of truth for n=k, gives us the resultthat Pn is also true for n=k+1.Hence by Principal of mathematical induction Pn is true.

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