If a + b − 1 = 1 + l n ( 2 a − 1 ) l n 4 + l n ( 2 b − 1 ) l n 4 then a = b?where a , b &gt; 0 are real numbers and ln is l o g e , then is a=b?

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If   a  +  b  −  1  =  1  +            l      n      (              2        a            −      1      )              l      n      4        +            l      n      (              2        b            −      1      )              l      n      4       then   a  =  b?where   a  ,  b  &amp;gt;  0 are real numbers and ln is   l  o      g    e  , then is a=b?

furniranizq · Accepted Answer

Let   f  (  x  )  =  x  −      log    4    ⁡      (          2              x        +        2              −    4    )  . Then the given relation is   f  (  a  )  +  f  (  b  )  =  0Now   f  (  x  ) has a minimum at   x  =  1, which is   f  (  1  )  =  0. So   f  (  x  )  &amp;gt;  0 for all other values of   x  &amp;gt;  0. Hence for the relation to be satisfied,   a  =  b  =  1Updated: to show the minimum without calculus, note that  f  (  x  )  ≥  0    ⟺    x  ≥      log    4    ⁡  (      2          x      +      2        −  4  )    ⟺        4          x        ≥      2          2      +      x        −  4    ⟺        y    2    ≥  4  y  −  4    ⟺    (  y  −  2      )    2    ≥  0 where   y  =      2    x  and this also     ⟹    f  (  x  )  =  0    ⟺        2    x    =  2    ⟺    x  =  1

If a + b − 1 = 1 + l n (

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