Solve exponential-polynomial equation Solve the equation in <mrow class="MJX-TeXAtom-ORD">

Wade Bullock 2022-07-02 Answered
Solve exponential-polynomial equation
Solve the equation in R
10 3 x log 10 x + x ( log 10 2 x 2 log 10 x ) = x 2 + 3 x
To 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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Answers (2)

Isla Klein
Answered 2022-07-03 Author has 12 answers
Clearly x > 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 ) = 0
Now note that if ( y 3 ) ( y + 1 ) > 0, both terms on the left are positive, hence the equation cannot have a solution. Similarly, if ( y 3 ) ( y + 1 ) < 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 }
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Raul Walker
Answered 2022-07-04 Author has 7 answers
that is not a hard question !
you can assume :
10 a = x
then your problem turn out to be :
x 2 + ( 3 a 2 + 2 a ) x x a 10 3 = 0
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