Test of Mathematics Solution Subjective 67 - Four Real Roots

This is a Test of Mathematics Solution Subjective 67 (from ISI Entrance). The book, Test of Mathematics at 10+2 Level is Published by East West Press. This problem book is indispensable for the preparation of I.S.I. B.Stat and B.Math Entrance.

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Problem

Describe the set of all real numbers x which satisfy 2 $ {\log_{{2x+3}^x}} $ <1.

Solution

2 $ {\log_{{2x+3}^x}} $ < 1

Now x< 0 is not in the domain of logarithm.

So x> 0.

Now as x>0           2x+3 > 1.

So $ {(2x+3)^a} $ > $ {(2x+3)^b} $ for a>b
So 2 $ {\log_{{2x+3}^x}} $ < 1 or $ {(2x+3)^{\frac{1}{2}}} $ > x
or 2x+3 > $ {x^2} $
or $ {x^2} $ -2x -3 < 0
or -1 < x < 3

But x > 0 so set of all real number

x is 0 < x < 3.