How to solve this equation with linear n as well as polynomial n? I am banging my head against the

aniawnua

aniawnua

Answered question

2022-05-20

How to solve this equation with linear n as well as polynomial n?
I am banging my head against the wall, but somehow I can't find a closed form solution to this equation in n:
229 , 244 + 58 , 044 n = 130 , 000 1.78 n
Obviously, if there was no n multiplied with 58,044, then this would be trivial (take a log and solve it). I am at a loss about how to solve this one though. Any hints will be appreciated. Please don't copy paste the Wolfram Alpha solution here, as I am interested in a pen-paper solution approach to this problem.

Answer & Explanation

Leonard Mahoney

Leonard Mahoney

Beginner2022-05-21Added 10 answers

One solution of the equation a + b n = c d n is given by:
n = a b W ( x ) l o g ( d )
with x = c d a / b log ( d / b ), W being the Lambert function.
For the values you gave to the parameters, this leads to n = 3.68139 and n = 1.56157.
As suggested in comments, a preliminary look at the graph of the function is a very good idea. It will allow you to see the number of roots and locate them. From these rough estimates, use Newton method for polishing the roots.

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