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.
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.