I've always wondered this, but I don't know if it's possible.
Is there a general formula for finding the fractional base that will equate two decimal numbers.
For example:
\begin{gather} 11_4 = 5\\ 12_{12} = 14 \end{gather}
Given an expression of the form $A_C = B$, how can we calculate the $C$ that makes the statement true?
For example, what is $C$ for $140000_C = 150000$?
This is equivalent to solving polynomial equations. For example $11_x=5$ is the same as $x+1=5$ so $x=4,$ while $12_x=14$ becomes $x+2=14$ so $x=12$.
It will not always have an integer base as an answer. Also if you see the largest digit of the left number is say $a$, the base $x$ must be at least $a+1$, giving another constraint. Your example $140000_x=150000$ is equivalent to $$x^5+4x^4=150000.$$ Because of the $4$ we know $x \ge 5$. Of course here we already know $x\ge 10$ which is too small. But when $x=11$ the left side is $219615$ which is already too big. So this particular problem has no solution.
In general once one sets up the unknown base side of the equation as a polynomial in $x$, one is looking for solutions for which $x$ exceeds the largest digit (now a coefficient of some power of $x$), and is a positive integer. For this a root finder would be the best way to go, given the difficulty of solving high degree polynomial equations.