I have a series of numbers, say 2, 3, 7 that sum 12. I want to increase these numbers so that they sum 62. I could multiply each number by 62/12 to achieve that. However, all numbers would be scaled equally linearly. I want instead that 7 be scaled more than 2- i.e. bigger numbers to be scaled more.
One way to achieve that would be to square each number. $2^2+3^2+7^2=4+9+49=62$
In general, given $a_i$ and an $n$ I want to find a $k$ so that:
$\sum a_i^k=n$
I haven't been able to isolate the $k$. Can it be done or should I resort to finding it numerically (which sounds easy). I've tried using exponentiation...
$e^{\sum a_i^k}=e^n$
$\prod e^{a_i^k} = e^n$
But I can't seem to find any useful manipulation with $a^{b^c}$.
Thoughts?