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Questions tagged [logarithms]

Questions related to real and complex logarithms.

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Image of a straight line under $w = Log(iz)$.

In detail, I have to find the image of the straight line parallel to the co-ordinate axes for the function $w = Log(iz)$. What I have tried is as follows (Since I am new to complex-analysis, I must be ...
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14 views

Parametric representation of the real branches $\operatorname{W_{0}},\operatorname{W_{-1}}$ of the Lambert W function

$\require{begingroup} \begingroup$ $\def\a{\alpha}\def\la{\ln{\a}}\def\e{\mathrm{e}}\def\W{\operatorname{W}}$ $\def\Wp{\operatorname{W_0}}\def\Wm{\operatorname{W_{-1}}}$ For $x\in(-\tfrac1{\mathrm{e}...
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1answer
27 views

How to show that the following equality is true with the help of lambert W function?

How to show that if $$y=x(2^{1/x}-1)\tag{1}$$ then $$x=-\frac{y\log(2)}{yW(-\frac{2^{-1/y}\log(2)}{y})+\log(2)}.\tag{2}$$ Where $\log$ is the natural logarithm and $W$ represent the lambert W function....
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Find the natural log of a number without a calculator [on hold]

Calculate it without using the calculator. Are there any quick methods or approximations? Log(50) base e
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1answer
40 views

Integrating (x+1)/x

A textbook way to integrate $\frac{a}{x}$ is $\int \frac{a}{x} dx = a\ln(x) $ However the answer to the question $\int \frac{x+1}{x} dx$ is not $(x+1)\ln(x)$ but is rather $\frac{x}{x} + \frac{1}{...
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1answer
20 views

Solutions of a logarithmic equation.

Equation from Question: $$x^{(\log_3(x))^2 + \log_3(x^4)-3}=3^{2\log_3(x)}$$ The question states to find the number of solutions and the sum of integral solutions. The correct answer ...
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1answer
36 views

How does this entropy equation simplify?

This is from the bok "Pattern Recognition and Machine Learning" By Bishop. I am having a hard time following the last step of this equation Where Stirlings approxaimation is subtituted for $\ln N!$ to ...
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1answer
50 views

Is there a difference between using ln() and log() for calculating the periodic return of an asset

I am a highschool student with a small but decent knowledge of stocks. For a maths project, I'm investigating the probabilities and correlations of stocks. I came across this formula for calculating ...
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2answers
27 views

Function design: “stateless” recursive asymptotic to 1

I'm trying to design a function with the following requirements: it will be implemented in an electronic device, where it will be called a discrete number of times. However, it will only be given the ...
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1answer
17 views

Calculating the base-2 logarithm given an n-bit normalized fractional number

I recently started reading Complex Digital Circuits by Jean-Pierre Deschamps and ran into a mathematical curiosity that has stalled me on making progress. For context, the author is describing the ...
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1answer
21 views

Equivalent values

$$\mathcal{O}(f (n)· L) = \mathcal{O}(f (n)\log n)$$ Where is $L$ is a depth of tree. How happened that $\ L = \log n $? Why this two values are equal?
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Help with an identity involving PolyLog and Logs.

I am plotting the following expression on Mathematica $$ \Im\left[-96 \text{Li}_3\left(e^{-i t}\right)-48 t^2 \log \left(1-e^{-i t}\right)\right]-96 \Re\left[t \text{Li}_2\left(e^{-i t}\...
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2answers
21 views

find the x-intercept of a natural logarithm function with a power of 2

find the x-intercept of the function y = ln(3x-2)^2. in order to find it, move the power 2 in front of the natural log. y = 2 ln(3x-2). for x -intercept, y = 0. Therefore, ln(3x-2) = 0. Hence, (3x-2)...
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1answer
22 views

Differential of sum of logs of probabilities

I'm trying to calculate the derivative with respect to $p1$, $p2$, ... $pk$ (each) of the following equation: $L = N_1 \log p_1 + N_2 \log p_2 + ... N_k \log p_k$ where $\Sigma_{i=1}^k p_i = 1$ i.e....
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1answer
34 views

Solving logarithm with variable in base and exponent for x

$$ log_{1+\frac{0.0231}{x}}{5x} = 10/9$$ What is the solution to this equation and how would I approach the problem? For the purposes of the question, algebraic manipulation with log and exponential ...
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1answer
50 views

Powers of complex numbers.

It is known that if $\alpha$ is a complex number, then for example, the equation $x^2 = \alpha$ has $2$ solutions. In general, there are $n$ values for the $n$-th roots of a number. In other words, if ...
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Deriving a multivariate inverse function

In my math assignment I have to find the inverse of $$ f(x_1, x_2) = \left(\ln \left(\frac{x_2}{x_1}\right), x_1^2 + x_2^2\right) $$ Now I already have looked into this, and came up with the ...
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0answers
29 views

Is there any formula or method to calculate logarithms with the following information available $y = a^{x}?$

I would like to know how calculators calculate log or how in ancient times, Mathematicians used to develop log sheets for specific variables. For instance, for $y = a^{x}$ we know that $$x =( y/a)-(a^...
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1answer
27 views

Branches of $\arg(z)$ and $\log(z)$

I do not understand this excerpt from my textbook: It says that any open disk G that excludes the origin, G has a branch. But in the problem, it says there is no such branch? I do not understand the ...
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0answers
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Logarithm transformation for a function in Machine Learning

I was looking for a clarification on why the function (the one raised to the power of M) was transformed to a logarithm function. I know it will be used to find p when L is maximised ( derivative of ...
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35 views

Solve For Variable Inside Natural Log and Outside

So, the equation I have is more simplified on here compared to the one I have to solve but here it is: $$10 = 2 \cdot \dfrac{x+4}{5x} \cdot \ln\left(\dfrac{2x}{6}\right)$$ I am unfamiliar with how ...
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1answer
24 views

how can i evaluate logarithmic series

If I have $\log_{21} 1 + \log_{21}3+\log_{21} 7+...+\log_{21} 441$ and all the values are divisors of 441. How would i evaluate this sum? I know the multiplication rule would make it $\log_{21} (1*...
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Voyage into the golden screen

We start from A004718 named "The Danish composer Per Nørgård's "infinity sequence", invented in an attempt to unify in a perfect way repetition and variation" $$a(2n) = -a(n), a(2n+1) = a(n) + 1, a(0)=...
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2answers
43 views

Why is it that $ \ln(1) = 0 $ but $ e^{(i2\pi)}=1$?

Why is it that $ e^{(i2\pi)} = 1 $ but $ \ln(1) = 0 $? In other words, is $2\pi i = 0$? I know that $e^0 = 1$, but should it also equal $e^{2\pi i}$?
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1answer
35 views

A logarithmic inequality.

Show that $$|\log (1-z)| \leq |z| + \frac {|z|^2} {(1-|z|)^2},$$ for all $z$ with $|z|<1.$ I know that $\log(1-z) = \log |1-z| + i\ \text {arg} (1-z).$ This shows that $$\begin{align*} |\log(1-...
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0answers
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Does a matrix logarithm expansion work when the identity matrix is rank deficient?

Apologies if this seems trivial to some, but I've been unable to find any decisive literature on this (physicist asking here). Consider the following Taylor expansion of the matrix logarithm: $log(...
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1answer
32 views

Is the Principal Logarithm Function Entire and Can Its Value Ever Be Less Than 0?

Define the Principal Logarithm as Follows: Log(z) = ln|z| + iArg(z) Where z is not equal to 0, z = r$e^{iθ}$, and θ = arg(z) and θ lies on (-π, π]. Is this function entire? And if so, can its value ...
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How to use Logarithm in the Sieving step of Quadratic Sieve technique

I am working on a program to factor large semi-prime numbers. I am using the simple Quadratic Sieve technique. My program works well but lot slower because during the sieving process (when I was ...
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1answer
27 views

Can $\frac{x-c}{x} = \frac{y-c}{y}e^{-\frac{(x/y)-1}{(x/y)+1}}$ be solved explicitly for $x$ and $y$

where $c>0$ and $x,y \geq 0$. Is there an explicit way (closed form) to solve the equation in the title? I believe the answer is that it must be that $x=y$, at least according to my logic below, ...
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3answers
30 views

Solving logarithmic equations algebraically

1. $\log_{10}(x+4) -\log_{10}x = \log_{10}(x+2)$ $10^{\log_{10}(x+4)} - 10^{\log_{10}x} = 10^{ \log_{10}(x+2)}$ $x+4 - x = x+2$ $x=2$ 2. $\ln(x+1)^2 = 2$ $e^{\ln(x+1)^2 } = e^{2}$ $(x+1)^2 = e^...
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1answer
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question on dividing an image multiple times

So for each of these images, I need to divide them four times, so that they are 4 times nested within the largest image. I understand that part. a) For the first figure (the L-shaped figure), how ...
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Intersection of two exponential graphs

Setting the two equations equal yields $3^{2x}-3^x = 4*3^x$ Let $y=3^x$ Then we have $y^2-y=4y$ $y(y-5)=0$ $y=0,5$ $3^x = 0, 3^x =5$ $x\log 3 =0, \implies x=0$ and $x\log 3 = 5 \implies x = \...
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3answers
64 views

How to know which value is bigger? [duplicate]

Which is bigger between $2018^{2019}$ or $\ 2019^{2018}\ $? When taking logs of both sides and I get: $2019\log(2018)\ $ and $\ 2018 \log(2019)$ I know $\log 2019\gt \log 2018$ so does this mean ...
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2answers
45 views

Function has only one solution

How can I prove that this has only one solution for 0< x <2 when f(x)=2? $$f(x)= \log_{k} (6x-3x^2)$$ When I try to solve this equation I get $$k^2=-3x(x-2)$$ But then I'm stuck as it feels ...
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4answers
212 views

Trans series for an integral

In doing a computation regarding the electroweak phase transition in the early universe, I came across the following integral: $$I=\int_0 ^ {\infty} x^2 \text{ln}(1-\text{exp}(-\sqrt{x^2+u^2}))\text{...
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3answers
31 views

How can I isolate a variable in a logarithmic equation?

Sorry for such a simple question. I have $y\ln(5) = 2\ln 3$ And I wanted to know if I can solve for $y$ by just dividing the logarithm on both sides? So I would get $y = \frac {2\ln 3}{\ln(5)} = ...
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3answers
39 views

Are there any points on the curve $y=\frac{x}{2}+\frac{1}{2x+4}$ where the slope is $\frac{-3}{2}$?

I am not sure how to find points of $y=\frac{x}{2}+\frac{1}{2x+4}$ where the slope is $\frac{-3}{2}$ without looking at a graph. I can simplify the function by writing it as $y=\frac{x(x+2)+1}{2(x+2)...
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2answers
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How to see that $1$ is a solution of $x^{x^2−3x} = x^2$

I tried to solve the problem below to get all the positive solutions: $$x^{x^2−3x} = x^2$$ By using $\ln$ on both sides, I get that one solution is $\displaystyle\frac{3 + \sqrt{17}}{2}$. But $1$ is ...
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1answer
30 views

What is a good estimate for this log sum?

Given $n\gg0$ what is a good estimate for $$\sum_{i=1}^K\log\Big(\frac{n}{3^i}\Big)?$$ I am particularly interested in case of $K=O(1)$ and $K=O((\log n)^c)$ at a fixed $c>0$.
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2answers
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Series that sums to ln2…but how? [closed]

I have this series: $$\sum_{n=1}^\infty \frac{1}{2^n \cdot n}$$ According to Wikipedia this series converges to $\ln2$, but I can't seem to prove this result. Any idea?
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1answer
54 views

Branches of log [closed]

I don't really understand what branches of log are in complex analysis. The definition I have is that 'a branch of log $z$ in G is a continuous function $l$ in G such that, for each $z$ in G, the ...
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1answer
26 views

Where is the error in the following line (complex logarithm)?

Let's take the principal branch $L(Re^{i\varphi})=\ln(R)+i\varphi$ of the logarithm. I got myself confused now over this: Where is the error in the line $$i\pi=L(e^{i\pi})=L(e^{i\pi}\cdot 1)=L(e^{i\pi}...
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1answer
49 views

Solving an inequality involving exponentials

For a homework problem, I'm supposed to compute the integral of $f(x) = x e^{-x^2}$, which I did, and got a result of $F(x) = -\frac{1}{2} e^{-x^2} + C$. Secondly, I'm supposed to find the smallest ...
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2answers
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questions on graphs of logarithms and exponentials

For this, if i take the log, I know that $\ln (4) > \ln(3)$ The question doesn't specify if $x>0$ but I will assume so. So I have $-x\ln(4) < -x\ln(3)$ So $g(x)=3^{-x}$ is the "bigger graph."...
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1answer
38 views

Difficulty deducing derivatives of particular summation formulas

I have difficulty concluding the derivatives of the following: $$y=\ln\bigg(\sum_{j=1}^ng_j\cdot e^{xE_j}\bigg)$$ and $$y=x\cdot \sum_{j=1}^n \bigg(g_j \cdot \frac{N}{\sum_{j=1}^ng_j\cdot e^{xE_j}}\...
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3answers
36 views

Solving Equations using logarithm [closed]

Here is a system of equations for which I am having difficulty solving: \begin{cases} a^{2x}.b^{3y}=m^5 \\ a^{3x}.b^{2y}=m^{10} \end{cases}
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0answers
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Bound on the Kolmogorov complexity of integers

I am reading Elements of Information Theory (Thomas M. Cover, Joy A. Thomas, 2nd edition) in which the following theorems are given (page 475--476): For any integer $n$ $$ K(n) \leq \log^* n + c. $$...
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1answer
30 views

Differentiation of logarithmic functions.

https://imgur.com/CwWNDts If $ y = \log_{10} X + \log_e 10 + \log_x X + \log_{10} 10$, then find $\frac{dy}{dx} $ So I was doing this question and when I got my result it didn't match with the ...
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1answer
31 views

Solve the system of equations with exponential term.

I am trying to solve this system of equations. I know the answer but I am struggling with the working. I need to solve for $m$ and $s^2$ in terms of all the other parameters. The system of equations ...
8
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2answers
1k views

Problem with estimating a sequence with intuition

I've frequently used "intuition" to solve limits at infinity. For example, if someone asked me what is: $$ \lim_{x \to \infty} f(x) = \frac {x^5 + x^3 + x}{x^2} $$ Or a sequence that can be ...