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How do we define domain for this log equation? Also, Please explain what conditions are to be taken when base is a variable in log inequalities?
125. log x–1 3 = 2.

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Please find below the solution to the asked query:

logAX is defined when X>0A>0A≠1logx-13=2logx-13 is defined when x-1>0 and x-1≠1⇒x>1 and x≠2 i.e. x∈1,2∪2,∞⇒3=x-12 ⇒x-1=±3⇒x=1±3But x=1-3 is not element of set 1,2∪2,∞hencx=1+3 Answer

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x - 1 greater than 0.
x greater than 1.

x - 1 is not equal to 1
x is not equal to 2

The above two conditions should be taken and then make two cases.
First case is when base lies between 0 to 1. It is a decreasing function and sign will change in this case while removing log.
Second case is when base lies between 1 to infinity. It is an increasing function and sign will not change while removing log.
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