An algebra problem by Prayas Rautray
Find the integer value of x satisfying 2 x = 4 x .
The answer is 4.
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1 solution
Notice that if x < 2 , then 2 x − 2 will never be an integer. For x > 4 , we can compute the derivatives of each function ( 2 x − 2 ln ( 2 ) and 1 ) respectively. This will show that 2 x − 2 never grows faster than x, as shown by the statement: when x = 4 , 4 > 1 . Additionally, as the derivative of 2 x is positive, it is a monotonically increasing function (for x > 4 to show that 2 x is always increasing, so it will never drop back down below x .
These two properties of 2 x show that for x > 4 , 2 x > x .
How do we know that this is the only integer solution to this problem?
We can rewrite the problem as 2 x = ( 2 2 ) ( x ) , or 2 x − 2 = x .
Since the answer (which includes both the LHS and RHS of the answer) are integers, the RHS x must be an integer. Since 2 i n t e g e r = a n o t h e r i n t e g e r , we can start from the case x − 2 = 0 , or x = 2 .
For x = 2 , the statement becomes 2 2 − 2 = 2 , which is obviously not true.
For x = 3 , we have the equality 2 3 − 2 = 3 , which is again false. Since the LHS increases exponentially, while the RHS increases linearly, we can check the next few values of x and see if there is any integer solution (as said in the problem statement).
Substituting x = 4 gives 2 4 − 2 = 4 , which is true. Therefore, by trial and improvement, x = 4 is an integer solution to this problem.