This problem is related to a time constant (herein called $x$) possessed by an article heated in a constant-temperature oven. Below I have simplified the presentation of the problem so as to concentrate on the math aspects. I do not want to get off onto discussing times and temperatures.
Is there a direct symbolic solution for $X$ in the following equation? i.e, $X=$ _________
(Even a converging series would be nice.)
$J\left(\exp\left(\frac ax\right) -1\right) = K\left(\exp\left(\frac bx\right) -1\right)$
Assume (for laboratory use):
$J, K, a, b$ are real;
$a > b > 0$;
$|K| > |J| $
In any given experiment, $K$ and $J$ have the same polarity (either both are positive, or both are negative)
One set of numerical values:
$J=18; K=52; a=3.8; b=2.25; x= 1.824551$
I converted the equation to an infinite series, but all of the $x$'s were in the denominator:
$$... + (Kb^n-Ja^n)/(x^nn!) + \ldots = 0 \;\;\;(n=1, 2, 3, \ldots)$$
I am not a mathematician. I came across this problem while working in a laboratory in 1983, and have worked on it off and on since then.