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EQUATION 1:

$\text{Constant} = \frac{\ln(y-x)}{\ln(y)}$

GIVEN:

  1. $0 <$ Constant $< 1$
  2. The $x$ and $y$ values are always positive. The closed form solution may ignore zero and negative $x$ and $y$ values.

SIMPLIFIED EQUATION 1:

Equation 1 may be simplified using log rules to $y-y^\text{Constant} = x $

The simplified equation is $x$ as a function of $y$. I would like to solve $y$ as a function of $x$. I am stuck at this point.

BACKGROUND: I upload a table of $x$ and $y$ values to legacy software. I am commonly told to change the constant. I am able to solve $y$ computationally using Excel. Using Excel introduces manual steps in my workflow when I am told to change the constant. For example, my current workflow is: change the constant, use Excel solver to solve Equation 1 using the updated constant for y as a function of $x$ (e.g., $x$ values $.01$ to $9,999.99$ at a step size of $.01$) , upload table with $x$ and $y$ values from excel into legacy software. The legacy software is able to handle closed form solutions. Thus my desire to automate the process of generating the table in the legacy software using the closed-form solution (i.e., without Excel).

QUESTION: Is there a closed-form solution to the equation $1$? If so, what is the closed-form solution?

Тyma Gaidash
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Graph121
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2 Answers2

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No, there is not. EDIT: Actually (As pointed out by a comment of Tyma Gaidash), there is in terms of obscure inverse function, see forumla 1 in this post. It even has an excel implementation.

There are closed-form solutions for specific values of $Constant$ though.

e.g. $Constant=\frac12$ gives the equation $y=(y-x)^2$ which can be solved to give $y=\frac{2x+1-\sqrt{4x+1}}2$.

Other values of $Constant$ with "closed form" solutions are $\frac14,\frac13,\frac23,\frac34$, although the formulas are very very unwieldy.

For example, the $\frac13$ case gives $$y = \frac{\sqrt[3]{\sqrt3 \sqrt{27 x^2 + 4} - 9 x}}{\sqrt[3]{18}} - \frac{\sqrt[3]{\frac23}}{\sqrt[3]{\sqrt3 \sqrt{27 x^2 + 4} - 9 x}} + x$$

It will thus be impossible to solve this equation in terms of standard functions.

EDIT2: Edited out false claim that $\frac12$ case has multiple solutions

student91
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  • You introduced a false solution by squaring. There is only one solution. (Try graphing $f(y)=y-y^c$ to see this.) – mr_e_man Mar 01 '23 at 20:59
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$$\frac{\ln(y-x)}{\ln(y)}=c$$

Your equation is an equation of elementary functions. It's an algebraic equation over the reals in dependence of $\ln(y-x)$ and $\ln(y)$. Because the terms $\ln(y-x),\ln(y)$ are algebraically independent, we don't know how to rearrange the equation for $y$ by only elementary operations (means elementary functions).
I don't know if the equation has solutions in the elementary numbers.
$\ $

$$\frac{\ln(y-x)}{\ln(y)}=c$$ $$\ln(y-x)=c\ln(y)$$ $$y-x=e^{c\ln(y)}$$ $$y-x=y^c$$ $$y^c-y+x=0$$

For rational $c$, this equation is related to an algebraic equation over $\mathbb{C}$ and we can use the known solution formulas and methods for algebraic equations.

For rational $c\neq 0,1$, the equation is related to a trinomial equation.

For real or complex $c\neq 0,1$, the equation is in a form similar to a trinomial equation. A closed-form solution can be obtained using confluent Fox-Wright Function $\ _1\Psi_1$ therefore, as is written in https://mathoverflow.net/questions/426543/is-it-possible-to-solve-for-y-in-this-equation.
$\ $

Szabó, P. G.: On the roots of the trinomial equation. Centr. Eur. J. Operat. Res. 18 (2010) (1) 97-104

Belkić, D.: All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells. J. Math. Chem. 57 (2019) 59-106

IV_
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