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Question

The number of solution(s) of 1+logx(4x10)=(log10log10n1)logx10 for a given value of n(1,104){103} is 


Solution

Given equation:
1+logx(4x10)=(log10log10n1)logx10
For the log to be defined, 
4x10>0;     x>0;  x1x<4x(0,4){1}(1)

Now,
1+logx(4x10)=(log10log10n1)logx101+logx(4x)logx10=logx10(log10log10n)logx10logx(x(4x))=logx10(log10log10n)log10(4xx2)=log10log10n4xx2=log10nx24x=log10n(x2)2=4log10nx=2±4log10n

When n(1,104){103}, so 
log10n(0,4){3}4log10n(0,4){1}4log10n(0,2){1}

For a given value of n(1,104){103}, we get two solutions as
x1=2+4log10nx2=24log10n

Mathematics

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