# 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:

$$229,244 + 58,044 \cdot n = 130,000 * 1.78^n$$

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.

• How about graphical solution? – Bernd Jan 15 '14 at 9:56
• Hi Bernd, that is where I am already at. I am more interested in a closed-form solution via some algebraic manipulation. – joshi Jan 15 '14 at 9:57
• – gammatester Jan 15 '14 at 9:58
• Hi there! I'm afraid this is a trascendental equation, so you cannot solve explicitly for $n$. As it was discussed here: math.stackexchange.com/questions/638399/…, you might consider a numerical approach for $n$ as well as using the Lambert $W$ function. Cheers. – Dmoreno Jan 15 '14 at 9:58
• Other than graphical it might work with Newton-Approximation – Bernd Jan 15 '14 at 10:00

## 1 Answer

One solution of the equation $\rm a + b n = c d^n$ is given by:

$$\rm n = - \frac ab - \frac{W(x)}{log(d)}$$

with $\rm x = - c d^{-a/b} \log(d/ b)$, $\rm W$ 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.