How can I prove that there is no closed form solution to this equation? $$2^x + 3^x = 10$$
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19A rigorous proof requires a rigorous definition of "closed form solution". – Bill Dubuque Nov 04 '10 at 18:33
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1Okay. Let say that you can use exponentials, logarithms, digits 0,...,9, variable $x$, $n$th roots, four elementary operations (+ – × ÷) and make compositions and combinations of them. The expression should contain only finitely many characters as written in LaTeX. In particular, symbols $\sum$, $\int$, $\cdots$, $\ldots$ are forbidden. – Jaska Nov 04 '10 at 18:51
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15@Jaska: in that case you might be interested in reading http://www.jstor.org/stable/2589148 . – Qiaochu Yuan Nov 04 '10 at 19:04
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@Qiaochu Yuan: Thanks for that! – Jaska Nov 04 '10 at 19:15
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I think that if you could find a function $g(t,u,v)$ such that $x_{0}=g(2,3,10)$, $f(x)=2^{x}+3^{x}-10$, $f(x_{0})=0$, then $2^{x}+3^{x}=10$ would have a closed form. My problem is that I am not able to prove there is no such $g$. – Américo Tavares Nov 04 '10 at 20:42
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5@Jaska: If you don't have JSTOR access, you can find the paper at http://arxiv.org/abs/math/9805045 – Ross Millikan Dec 15 '10 at 04:59
4 Answers
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A closed-form solution is a solution that can be expressed as a closed-form expression.
A mathematical expression is a closed-form expression iff it contains only finite numbers of only constants, explicit functions, operations and/or variables.
Sensefully, all the constants, functions and operations in a given closed-form expression should be from allowed sets.
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The following parts of the answer are only for closed-form solutions that are expressions of elementary functions. According to Liouville and Ritt, the elementary functions can be represented in a finite number of steps by performing only algebraic operations and / or taking exponentials and / or logarithms.
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$2^x+3^x=10$ is a transcendental equation: $e^{\ln(2)x}+e^{\ln(3)x}=10$. The left-hand side of this equation is the functional term of an elementary function. Because $\ln(2)$ and $\ln(3)$ are linearly independent over $\mathbb{Q}$, the expressions $2^x$ and $3^x$ are algebraically independent: MathStackExchange: Algebraic independence of functions. Therefore one can prove with help of the theorem of [Ritt 1925] (which is also proved in [Risch 1979]) that a function $x\mapsto 2^x+3^x$ cannot have a partial inverse over an open domain $D\subseteq\mathbb{C}$ that is an elementary function. It is not possible therefore to rearrange the equation according to $x$ only by applying only elementary operations (elementary functions) one can read from the equation.
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The question of solvability of some special kinds of equations by elementary functions is treated in [Rosenlicht 1969].
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$2^x=e^{\ln(2)x}$, $3^x=e^{\ln(3)x}$
$2^x$ and $3^x$ are $\begin{cases} \text{algebraic}&\text{if }x\text{ is rational}\\ \text{transcendental}&\text{if }x\text{ is algebraic and irrational}\\ &\text{(Gelfond-Schneider theorem)}\\ \text{transcendental}&\text{otherwise(?)} \end{cases}$
Let $x_0$ be a solution of your equation. If $x_0$ is not rational, $2^{x_0}$ and $3^{x_0}$ are algebraically independent: MathStackExchange: Algebraic independence of functions. That means, $2^{x_0}$ and $3^{x_0}$ cannot fulfill together an algebraic equation, in particular not the equation $2^{x_0}+3^{x_0}=10$. That means, the equation can have only rational solutions.
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The existence or non-existence of elementary solutions (that are elementary numbers) could possibly be proved by the methods of [Lin 1983] and [Chow 1999]. But both methods need the Schanuel conjecture that is unproven.
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[Chow 1999] Chow, T. Y.: What is a Closed-Form Number? Amer. Math. Monthly 106 (1999) (5) 440-448 or https://arxiv.org/abs/math/9805045
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By the way, your equation can be written in form
$$H^{(-x)}_3=11$$
where H is the generalized harmonic number: https://www.wolframalpha.com/input?i=HarmonicNumber%5B3%2C+-x%5D%3D11
So to find $x$, you should investigate the inverse function of generalized harmonic number.
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everybody. I am not a mathematician. On the last 2 years, however, I have been working on a new function that is a modification of Lambert. You know, it is called Lambert-Tsallis function, originally published at https://doi.org/10.1016/j.physa.2019.03.046, $W_q(z)$. Actually, it is a multivalued function (depending on $q$ value) and it is the solution of a polynomial equation
$ W_q(z) \cdot \bigg( 1 + (1-q)\cdot W_q(z) \bigg)^\frac{1}{1-q} = z$
$x$ can be written in a closed-formula using $W_q(z)$, in 2 different ways. To tell you the truth it could be 3 different ways, but the last I left as an exercise. It does not use series expansion, any order special function, and no approximation. It is exact. It has been used to solve Fermat Equation and generalizations, also.
If one considers $u=log(2)$ and $v=log(3)$ we have
$x_1= \frac{1}{r\cdot u}\cdot log \bigg( \frac{1}{r} \cdot W_{\frac{r-1}{r}} \big( r\cdot 10^r \big) \bigg)$ with $r=(v-u)/u$ or
$x_2= \frac{1}{r\cdot u}\cdot log \bigg( \frac{1}{r} \cdot W_{\frac{r-1}{r}} \big( r\cdot 10^r \big) \bigg)$ with $r=(u-v)/v$
\begin{matrix} r & z & W_q(z) & x\\ +0.58496 & +2.2495 & (+1.17933 , +0.00000i) & (+1.72926 , +0.00000i) \\ -0.36907 & -0.1578 & (-0.18306 , +0.00000i) & (+1.72926 , +0.00000i) \end{matrix}
K Z Nobrega
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If a solution is in closed form, depends on the symbols and functions you allow to be a closed form.
The equation is related to an equation similar to a trinomial equation.
$$2^x+3^x=10$$ $$e^{\ln(2)x}+e^{\ln(3)x}=10$$ $x\to\frac{\ln(t)}{\ln(2)}$: $$e^{\ln(t)}+e^\frac{\ln(3)\ln(t)}{\ln(2)}=10$$ $$t+t^\frac{\ln(3)}{\ln(2)}=10$$ $$t+t^\frac{\ln(3)}{\ln(2)}-10=0$$ $$\frac{1}{10}t+\frac{1}{10}t^\frac{\ln(3)}{\ln(2)}-1=0$$ $t\to 10u$: $$u+10^{\frac{\ln(3)-\ln(2)}{\ln(2)}}u^\frac{\ln(3)}{\ln(2)}-1=0$$ $10^{\frac{\ln(3)-\ln(2)}{\ln(2)}}=-y,\frac{\ln(3)}{\ln(2)}=\alpha$: $$u-yu^\alpha-1=0$$
Now the equation is in the form of equation 8.1 of [Belkic 2019]. Solutions in terms of Bell polynomials, Pochhammer symbols or confluent Fox-Wright Function $\ _1\Psi_1$ can be obtained therefore.
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