Apologies in advance for any misused terminology, or if this is the wrong place for the question (I think it's okay though).
I am given a group of logarithmic expressions such as:
- $- (a \log(a) + (1 - a)\log(\frac{1-a}{2}))$
- $- (b \log(b) + (1 - b)\log(\frac{1-b}{4}))$
- $- (c \log(c) + (1 - c)\log(\frac{1-c}{5}))$
My goal is to find values of $a$, $b$, and $c$ that make these expressions equal.
The above is just an example of the sort of expressions I get, but they are always of the form $-(x\log(x) + (1-x) log(\frac{1-x}{n}))$, where $n$ is a an integer $\ge 1$, and $x$ is some symbol that doesn't occur in more than one expression in the group.
Ultimately I'll be using a computer to solve these problems. But my question is: Can I solve this by hand? And if so, how? Alternatively, can I distill the problem down to something simpler to code up?
I imagine I'd start by creating a set of equations that expresses equality between the expressions. So between expression 1 and 2 I made:
$- (a \log(a) + (1 - a)\log(\frac{1-a}{2})) = - (b \log(b) + (1 - b)\log(\frac{1-b}{4}))$
But I'm not sure how to go about solving for the values of the variables. I found a bunch of info on solving systems of linear equations but I'm not sure what to do with the logs.
I wrote this when the expressions in the question didn't have leading negatives. However, I think the method still works in the same way.
This solution requires computation that cannot be done (or at least would be very tedious) by hand.
Let's simplify the general form of the expressions you're setting equal to each other:
$$x \log(x) - \left(1-x\right))\log\left(\frac{1-x}{n}\right)$$
$$x \log(x) - \left(1-x\right)\left(\log\left(1-x\right)-\log(n)\right)$$
Note that for both $\log(x)$ and $\log(1-x)$ to be real numbers, $x$ must satisfy $0 < x < 1$. We'll continue with this assumption.
$$x\log(x) - \left(1-x\right)\log\left(1-x\right) + \left(1-x\right)\log\left(n\right)$$
Set this equal to a constant $k$:
$$x\log(x) - \left(1-x\right)\log\left(1-x\right) + \left(1-x\right)\log\left(n\right) = k$$
$$x\log(x) - \left(1-x\right)\log\left(1-x\right) = \left(x-1\right)\log\left(n\right)+k\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(1)$$
This is a graph (from Wolfram Alpha) of the left side of the above equation, strictly between $x=0$ and $x=1$:
We want to find a fixed value of $k$ for which there exist values of $x$, call them $x_n$ (in your question, these were the $a$, $b$, $c$, etc.) for every integer $n \geq 1$, so that equation $(1)$ holds.
In particular, this value of $k$ must yield a solution when $n=1$. When $n=1$, $\log(n) = \log(1) = 0$, so $\left(x-1\right)\log\left(n\right)+k$ is just the horizontal line given by $y=k$. This means $k$ must be between approximately $-0.14$ and $0.14$ in order for there to be a solution to equation $(1)$.
When $n>1$, $y=(x-1)\log(n) + k$ is a line with positive slope that intersects the $y$-axis at $y=k-\log(n)$.
I propose that $k=0.1$ yields a solution $x_n$ for all integers $n \geq 1$.
This is a graph of the right side of equation $(1)$, superimposed on the graph of the left side, when $n = 2$ and $k = 0.1$:
Note that the line $y=(x-1)\log(n) + 0.1$ intersects the vertical line $x=1$ at $y=0.1$, regardless of what we choose $n$ to be.
My graphical skills are poor, so you'll have to imagine the set of lines $y=(x-1)\log(n) + 0.1$ over all integers $n \geq 1$ as a set of lines that all have non-negative slope and all pass through the point $(x,y) = (1,0.1)$.
It should be clear that there must be an $x_n$ that solves equation $(1)$ for each $n$.
The value we chose for $k$, $0.1$, was certainly not the only one. Any $k$ in some neighborhood of $k=0.1$ would yield a set of $x_i$ that work with equation $(1)$.
This implies that the solutions to your infinite system of equations are nowhere close to unique. When setting all of the equations equal to each other, I would recommend choosing a value of $k$ to set them all equal to, e.g. $k=0.1$.
Finding the values of $x_n$, once you've fixed a $k$ shouldn't be too difficult to code. For each $n$, finding $x_n$ is finding the intersection of a line that goes through $(x,y) = (1, k)$ with slope $\log(n)$, with the curve $y=x\log(x) - \left(1-x\right)\log\left(1-x\right)$.
I hope that helps.