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Question

If logx5=a and log20x=b then find the value of logx10.


Solution

we know log a to base b is loga/logb
so from que we can write
a=(log5/logx)
b=(logx/log20)
therefore 1/b=(log20/logx)
a+(1/b)=(log5+log20)/logx
(ab+1)/b=log(100)/logx               //loga+logb=log(ab)
(ab+1)/b=2log(10)/logx               //log(x^2)=2logx
taking 2 to lhs
(ab+1)/2b=log10/logx
that is
log10 to base x=(ab+1)/2b.....

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