Solve exponential-polynomial equationSolve the equation in R 10 − 3 x log 10 ⁡ x + x ( log 10 2 ⁡ x − 2 log 10 ⁡ x ) = x 2 + 3 xTo be fair I wasn't able to make any progress. I tried using substitution for the logarithms, but it didn't help at all.This is a contest problem, so there should be a nice solution.Any help? Clue?

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Solve exponential-polynomial equationSolve the equation in       R        10          −      3            x                  log                  10                    ⁡      x        +  x  (      log          10        2    ⁡  x  −  2      log          10        ⁡  x  )  =      x    2    +  3  xTo be fair I wasn&#039;t able to make any progress. I tried using substitution for the logarithms, but it didn&#039;t help at all.This is a contest problem, so there should be a nice solution.Any help? Clue?

Isla Klein · Accepted Answer

Clearly   x  &amp;gt;  0. Let   x  =      10    y  . Then we have       10          −      3            x    y    +  x  (      y    2    −  2  y  )  =      x    2    +  3  x    ⟺        10          −      3            x          y      −      1        +  (  y  −  3  )  (  y  +  1  )  =  x    ⟺        10                  y        2            −      y      −      3        +  (  y  −  3  )  (  y  +  1  )  =      10    y      ⟺                                10          y                (                  10                      (            y            −            3            )            (            y            +            1            )                          −        1        )            ⏟        +                    (        y        −        3        )        (        y        +        1        )            ⏟        =  0Now note that if   (  y  −  3  )  (  y  +  1  )  &amp;gt;  0, both terms on the left are positive, hence the equation cannot have a solution. Similarly, if   (  y  −  3  )  (  y  +  1  )  &amp;lt;  0, both terms are negative and again we cannot have a solution. Hence the only solutions are when   (  y  −  3  )  (  y  +  1  )  =  0    ⟹    y  ∈  {  −  1  ,  3  }    ⟹    x  ∈  {      1    10    ,  1000  }

Raul Walker · Accepted Answer

that is not a hard question !you can assume :      10          a        =  xthen your problem turn out to be :      x    2    +  (  3  −      a    2    +  2  a  )  x  −            x      a              10      3        =  0

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