# Tagged Questions

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### Why doesn't Logz/z have zeros?

Our book claims that $\frac {Logz}{z}$ has no zeros, where Logz is the principle branch of the complex natural logarithm. However, $Logz=log|z|+iArg(z)$, correct? So $Log1=log|1|+iArg(1)=0+i0=0.$ ...
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### Find the order of magnitude of the equation solution

Find the order of magnitude of the following equation solution: $$x(\ln x)^{2001}=n$$
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### Need help with a proof concerning zero-free holomorphic functions.

Suppose $f(z)$ is holomorphic and zero-free in a simply connected domain, and that $\exists g(z)$ for which $f(z) =$ exp$(g(z))$. The question I am answering is the following: Let $t\neq 0$ be a ...
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### What is the domain of this function? (Don't know how to solve it, logarithms…)

Please explain how you solved it, thanks. $f(x)=\sqrt{\log_x2 - \log_2x}$
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### Solve exponential-polynomial equation

Solve the equation in $\mathbb{R}$ $$10^{-3}x^{\log_{10}x} + x(\log_{10}^2x - 2\log_{10} x) = x^2 + 3x$$ To be fair I wasn't able to make any progress. I tried using substitution for the ...
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### Only root of a sum?

I have the following equation: $\sum\limits_{i=1}^{k}\left(n_{i}-n\cdot p_{i}\right)\log p_{i}=0$ where $p_{1},p_{2},...,p_{k}$ are the unknown variables with the condition: ...
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### Solving a logarithmic polynomial

I want to solve this equation for $x$: $${\frac{1}{\sqrt{2 \pi x}} \left(\frac{e z}{2x}\right)^x} = \epsilon$$ Is there a closed form for it, or does it have to be solved numerically? I can turn it ...
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### At which parameter value $c>0$ do the number of solutions of $\log(1+x^2)=x^c$ change?

I'm looking at the functions $x\mapsto \log(1+x^2)$ and $x\mapsto x^c,\ c>0$ on the interval $\mathbb R^+_0$. I'm interested in the properties of $$\log(1+x^2)=x^c.$$ Graphically, for small $c$, ...
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### Question about fixpoints and zero's on the complex plane.

Define property $A$ for an entire function $f(z)$ as $1)$ $f(z)=0$ has exactly one solution being $z=0$ $2)$ $f(z)=z$ has exactly one solution $=>z=0$ (follows from $1)$ ) $3)$ $f(z)$ is not a ...
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### How do I solve $\; 3^{2x+1}-10\cdot 3^x+3=0 \quad?$

Solve the following equation for $x$ : $\quad3^{2x+1}-10\cdot 3^x+3=0$ I am baffled to solve this equation. With graphing I have found the answers to be x=1 and x=-1. I would like to know how ...
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### Solving a transcendental equation consisting of a quadratic part and a part involving inverse Lambert W functions

Question statement I would like to solve the following equation in the two variables $x$ and $y$: \begin{gather} 0 = x^2 - a y^2 + i b [x y - W^{-1}(x)W^{-1}(y)] , \end{gather} where $a$ and $b$ are ...
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### Understanding accuracy of Newton's Method

In a numerical analysis book I'm reading it says that using the Newton error formula we can find an expression for the number of correct digits in an approximation using Newton's Method. Here's the ...
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### How do I cube/square a logarithm?

Btw, please don't give me the answer. I just wanna know how to raise a logarithm to its cube cause I'm stuck in this part, but don't solve it for me. $$\log \sqrt[3]x = \sqrt[3]{\log x}$$ I tried ...
### How to find the roots of $f(x)= \ln( \frac{x+1 }{x-2})$?
I can't solve this equation: $$\ln\left(\frac{x+1}{x-2}\right) = 0.$$ I do: \begin{align*} \ln \left( \frac{x+1}{x-2} \right)&=0\\ \frac{x+1}{x-2} &= 1 \\ x+1&=x-2 \\ ...
I need to find all the solutions to the following using logarithms: $(e^z-1)^3=1$ where z is a complex number. I am told that using roots of unity I can break this equation down but I must be missing ...