# Questions tagged [transcendental-equations]

Transcendental equations are equations containing transcendental functions, i.e. functions which are not algebraic. An algebraic function is a function that satisfies a polynomial equation whose terms are themselves polynomials with rational coefficients.

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### Are there any complex solutions to the equation ${x}^{2^x} = {2}^{{x}^{{x}^{2}}}$? [closed]

Thought of this question after learning about the Lambert W function and wanted to challenge myself. Are there any complex solutions to the equation $${x}^{2^x} = {2}^{{x}^{{x}^{2}}}$$ Tried to work ...
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### Finding $a$ such that $\tanh(x)-a\sin^2(x)=0$ has a double root

Given $f(x) = \tanh(x)-a\sin^2(x)$, what is the value of $a$ for which $f(x) = 0$ has a double root, and what is the value of that double root? My Work : Using MAPLE , I plotted a few graphs and ...
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### Simplifying a complicated transcendental equation

I want to solve for $x$ the equation $$x^{\alpha} (1-x)^{1-\alpha} = (\gamma x)^{\beta} (1-\gamma x)^{1-\beta}$$ where $\alpha,\beta,\gamma, x$ are all strictly between zero and one. If I'm not ...
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### What is the exact value for the solution to $(\sin x)^{\cos x}=2$? [duplicate]

What is the exact value for the solution to this equation? $$(\sin x)^{\cos x}=2 \tag1$$ A similar equation $$(\sin x)^{\sin x}=2 \tag2$$ can be solved using the Lambert's $W$ function. But with $(1)$,...
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### How would you solve $3^x = 2x + 3$ using the Lambert $W$ function

Could someone provide a solution to the equation $$3^x = 2x+3.$$ Our teacher told us to solve it graphically, but I was curious what the exact answers might be and just plugged it into Wolfram ...
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### Solve $x = -(e^{Bx}z-y)Be^{Bx} z$ for $x$ or show existence or uniqueness of solution.

Let $B,y,z \in \mathbb R$. Is it possible to solve the equation \begin{align} x = -(e^{Bx}z-y)Be^{Bx} z \end{align} for $x \in \mathbb R$? Or can one show that a unique solution $x \in \mathbb R$ ...
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### Solving $\frac{8^x-2^x}{6^x-3^x}=2$

$$\dfrac{8^x-2^x}{6^x-3^x}=2$$ It is easy to see that in the domain of $\mathbb{R}\setminus\{0\}$, the solution is $x=1$. https://www.desmos.com/calculator/dsei8j2sdq. Desmos adds that the only one. ...
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### How can we find Lambert W solution to $\dfrac {x\ln x}{\ln x+1}=\dfrac{e}{2}$?

Find all real solutions: $$\frac {x\ln x}{\ln x+1}=\frac{e}{2}$$ Cross multiplication gives $$2x\ln x=\ln (x^e)+e$$ I didn't see any useful thing here. I tried solving this equation in WA. The ...
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### Exact solution to the the equation $(2 \pi - \theta)\cos \theta + \sin \theta = 0$

I have been trying to solve the following goat grazing problem: A goat is tied to the outside of a circular fence. If the length of the rope is the same as the circumference of the fence, what is the ...
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### Find $x$ in the exponential equation $3^x+4^x+5^x=6^x$ [duplicate]

$3^x+4^x+5^x=6^x(R:x=3)$ I try but I can't finish $3^x+2^x~2^x+5^x=3^x.2^x$ $3^x\cdot2^x-3^x-2^x\cdot2^x=5^x$
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### Closed form of $\int_0^{\pi\text{ or }\frac\pi2}\cos(w (\cos(t)+a t-b))dt$.

Although an integral for $x=\dots$ exists, it is slightly harder to integrate. Dirac $\delta(t)$ helps solve $\cos(x)+ax=b$: $$\frac1{\sin(x)-a}=\int_a^b \delta(\cos(t)+at-b)dt\tag1$$ From numerical ...
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### On the Hurwitz stability of quasipolynomials

Suppose that $p,q\in\mathbb{R}^{+}$ and $a\in\mathbb{R}$. Consider the transcendental polynomial $$p\lambda^2+p\lambda-a\left(e^{p\lambda}-1\right)e^{-q\lambda}=0,\;\lambda\in\mathbb{C}.$$ I would ...
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### Is there a way to simplify $a\sqrt{1-a^2} + \arcsin(a) = \pi/4$?

A while ago, I was eating pizza and wondered that if you were to cut parallel to one of the radii, how far along would you need to cut in order to split a slice's area in half? In attempting to find a ...
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### Find the all explicit real roots of $x^{1/x}\ln x^2=x.$

Using standard mathematical functions, find the all real roots of the equation: $$x^{1/x}\ln x^2=x.$$ I saw this question in the group of students studying mathematics. I tried to solve the equation ...
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### Is the number satisfying $\eta=\sin(\cos(\eta))$ transcendental?

I was graphing the function $\sin(\cos(\sin(\cos(\sin(\cos...$ when I realized it started to flatten out. This meant that this approaches a constant. Since the sine and cosine repeat, we can make a ...
1 vote
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### Generalised Lambert W and irreducible polynomials

I want to find the root of a function $f$ defined as $$f(x)= e^{-cx} - \frac{P_n(x)}{Q_m(x)}$$ where $x,c$ are real numbers and $P_n,Q_m$ are irreducible polynomials of rank $n$ and $m$ respectively, ...
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### Cartesian equation for a transcendental / trigonometric curve

Hello! Please see figure above. I am searching for the cartesian equation for the curve in green, similar to how the equation for a semicircle is $f(x) = √(1 - x^2)$. I'm not sure if this is even ...
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### Help Me Minimize Freezer Burn on Ice Cream by Understanding Transcendental Equations

I'm a high school math teacher trying to stay engaged with the subject, and I've started wondering about this question: "Given a cylindrical ice cream container with radius r and height h, ...
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### How to extract $x$ from the exponential equation $(x-1)\cdot\left(2^{1/x}-1\right)=k$ when $k\in(0,1)$ and $x\gt 1$?

I spent quite a time searching the internet to find a way to extract $x$ from the exponential equation $(x-1)\cdot\left(2^{1/x}-1\right)=k$ when $k\in(0,1)$ and $x\gt 1$ I would appreciate any hints.
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### Solving $a x = [\ln(x) - b]^c$
I'm trying to solve the nonlinear equation $a x = [\ln(x) - b]^c$ for x, where a, b, and c are constants, for a project. I've tried numerical techniques like in Excel Solver, but the solutions seem ...