Questions related to real and complex logarithms.

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12
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168 views

A conjectured identity for tetralogarithms $\operatorname{Li}_4$

I experimentally discovered (using PSLQ) the following conjectured tetralogarithm identity: $$\begin{align}&\phantom{+\;}19\!\;\pi^4-570\ln^42-90\ln^43\\ &+720\ln^32\cdot\ln3+360\ln2\cdot\ln^...
11
votes
0answers
228 views

Relations connecting values of the polylogarithm $\operatorname{Li}_n$ at rational points

The polylogarithm is defined by the series $$\operatorname{Li}_n(x)=\sum_{k=1}^\infty\frac{x^k}{k^n}.$$ There are relations connecting values of the polylogarithm at certain rational points in the ...
11
votes
0answers
615 views

Understanding Ramanujan's approach in his proof of Bertrand's Postulate

I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of $\varphi(2x) - \varphi(x)$ What would be wrong with this approach for ...
9
votes
0answers
370 views

Two (strictly related) proofs by induction of inequalities.

This is a question I originally asked on MSE, receiving no answer, even with a bounty (which expired) on it. Therefore I am crosslinking in order to prevent duplication of effort: see here for the ...
8
votes
0answers
407 views

${\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos x)} \:\mathrm{d}x$ and $\int_{0}^{\pi/2} \frac{\log \cos x}{x^2}\:\mathrm{d}x$

I have found the following new result connecting two rational log-cosine integrals. Proposition. \begin{align} \displaystyle & {\mathfrak{I}} \int_{0}^{\pi/2} \frac{x^2}{x^2+\log ^2(-2\cos ...
6
votes
0answers
91 views

Trilogarithm $\operatorname{Li}_3(z)$ and the imaginary golden ratio $i\,\phi$

I experimentally discovered the following conjectures: $$\Re\Big[1800\operatorname{Li}_3(i\,\phi)-24\operatorname{Li}_3\left(i\,\phi^5\right)\Big]\stackrel{\color{gray}?}=100\ln^3\phi-47\,\pi^2\ln\phi-...
6
votes
0answers
172 views

Are known these identities, that I've deduce using Mobius inversion formula?

I would to know if this formula is right and know (these formula are the same by exponentiation), since I deduce this easily by a standar way (perhaps there are mistakes) using Mobius inversion from $$...
6
votes
0answers
100 views

The diferential equation $y' = \frac{\ln(x^2+y^2)}{x^2 + y^2}$

In my University, the integral calculus teacher gave me this diferential equation to solve. $$ y' = \frac{\ln(x^2+y^2)}{x^2 + y^2} $$ I dont have any clue of what form has the solution of this ...
5
votes
0answers
92 views

An integral to prove that $\log(2n+1) \ge H_n$

Dalzell integral The equation $$ \int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$ proves that $\frac{22}{7}-\pi>0$ because the integrand is positive. Some Dalzell-type integrals for $\log(...
5
votes
0answers
1k views

Choosing the branch of a logarithm

The problem: I am integrating complex logarithms over an angle $\phi$ over $[0,2\pi]$. It is quite complex (pun not intended) and I called Mathematica in to aid me. I am calculating an energy of a ...
4
votes
0answers
97 views

Trying to show that $\ln(x!) - \ln(\lfloor\frac{x}{2}\rfloor!) - \ln(\lfloor\frac{x}{3}\rfloor!) - \ln(\lfloor\frac{x}{6}\rfloor!) \ge \psi(x)$

I've been told that the approach below will not work. I would be interested if someone could help me to understand what will go wrong. Let: $$\psi(x) = \sum\limits_{p^k \le x} \ln p$$ So that (see ...
4
votes
0answers
117 views

Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong?

I've been reading through Jitsuro Nagura's classic proof that there is a prime between $x$ and $\frac{6x}{5}$ and it seems to me that it should be possible to improve on his upper bound for the second ...
4
votes
0answers
520 views

Logarithm of a complex number as intersections of two logarithmic spirals

In Penrose's book "The Road to Reality" page 97 figure 5.9 he shows the values of the complex logarithm in a diagram as the intersection of two logarithmic spirals. Can you please explain how these ...
3
votes
0answers
55 views

All solutions of $ z^i = i^z $

In the simple equation $ z^i = i^z $ how are all complex values found? $ z= \pm \, i, $ and what else? It can be found by inspection, but to find general solution: We take logs, there is a ...
3
votes
0answers
58 views

Want to know what's wrong?

I take a exercise from apostol's book. I was trying the next exercise and do it, but the answer (from the book) is different, and I don't know what part of my procedure it's wrong?. So I want to know ...
3
votes
0answers
58 views

Branch points and Riemann surfaces (analytic continuation),

Take probably the most typical example: $$f(z) = \sqrt{1-z^2}$$ This function uses the (complex) logarithm to define it: $$e^{\large \frac{1}{2}log(1-z^2)}$$ $$e^{\large \frac{1}{2}[ln|1-z^2| + ...
3
votes
0answers
15 views

optimal hill “rank” cannot be solved?

Okay so I was thinking about the following problem today: We have a guy who is h tall stand upon a paraboloid shaped hill of the form $z=-ar^n$ How far away (in r) does his friend who is also h ...
3
votes
0answers
54 views

Checking whether sequence $ x_n = \ln(n^2 + 1) - \ln(n) $ converges or diverges

I have to show whether $$ x_n = \ln(n^2 + 1) - \ln(n) $$ converges or diverges. I can write $$ x_n = \ln(n^2 + 1) - \ln(n) = \ln\left(\frac{n^2+1}{n}\right) = \ln\left(n + \frac{1}{n}\right). $$ ...
3
votes
0answers
81 views

How does $\ln(x)$ blow up at $0$ and $\infty$.

In general: How do I figure out how fast a function blows up at a certain point or infinity? How fast does $\ln x$ blow up at $0$? Does it blow up as fast as $1/x$, $1/x^2$, or maybe faster than any $...
3
votes
0answers
61 views

Is the product rule for logarithms an if-and-only-if statement?

If a function $f(x)$ is proportional to $\ln x$, then we know $$ f(xy) = f(x) + f(y). $$ My question is, Is the converse true? If we know that, for an unknown function f, $$ f(xy) = f(x) + f(y), $$ ...
3
votes
0answers
215 views

Solving an exponential equation without the quadratic formula

High school math student here. In my homework I was asked to solve $16^x +4^{x+1} - 3= 0$ and I used substitution to get $x=\log_4{(-2+\sqrt7)}$. However, this was in the chapter on ...
3
votes
0answers
83 views

Inverse of $x^2+\log^2\cos x$

I'm looking for the inverse of $$f(x)=x^2+(\log\cos x)^2$$ Where $f$ is defined from $[0,\pi/2)$ It dosen't have to be closed form, a sum, an integral or some special functions would be of interest ...
3
votes
0answers
49 views

Exercise concerning logarithms…

I have such a problem: find all the values of real parameter "a", for which the following inequality is true for any "x" that belongs to R. I will show you my solution, and please can you verify ...
3
votes
0answers
112 views

Fourier transform of a logarithm

How can one go about computing the 2d (or 1d, in either variable) Fourier transform of the function $$\ln(w^2-k^2)?$$
3
votes
0answers
93 views

Phrase and symbol for “geometric absolute value”$ e^{|\ln(x)|}?$

I'm calculate the median fractional difference between two vectors (to characterise the error in a quantity with a high dynamic range). If $a/b = 0.1$, the fractional difference is $10$, and if $a/b =...
3
votes
0answers
55 views

How to solve the equation $ (x-2)^{\log_{100}(x-2)}+\log_{10}(x-2)^5-12 = 10^{2\log_{10}(x-2)}$?

If $\displaystyle (x-2)^{\log_{10^2}(x-2)}+\log_{10}(x-2)^5-12 = 10^{2.\log_{10}(x-2)}$, then value of $x$ is ... My try:: Let $\log_{10}(x-2) = y\Leftrightarrow (x-2)=10^y$. Then $(10)^{y.\frac{1}{...
2
votes
0answers
28 views

Why is $\log z = \ln r + i\theta$ ($r>0, \alpha <\theta < \alpha + 2\pi$) discontinuous at $\alpha$?

In one book on complex variables it is written that, given the function $\log z = \ln r + i\theta$ (for proper citation, let's call it function (2), as in the book) ($r>0, \alpha <\theta < \...
2
votes
0answers
26 views

An Integral Substitution for $\int_0^{1} dy \left(\frac{M^2(y)}{\mu^2}\right)^{-\epsilon}$

I have integral (1) as a result from an advanced QFT problem. $$ \tag{1} I= \frac{\alpha}{2\epsilon} \int_0^1 dy \left( \frac{M^2}{\mu^2} \right)^{-\epsilon} + \mathcal{O}(\epsilon) $$ ...
2
votes
0answers
49 views

Bounds on twin prime counting function

I read somewhere (unfortunately I cannot find the paper again) that the twin prime counting function $\pi_2(x)$ satisfies $\pi_2(x) \leq C\frac{x}{\log^2x}$ for some constant $C$. How would one prove (...
2
votes
0answers
41 views

Solving the logarithmic rational equation

I'm wondering there exist the way to solve the equation form of: $$ \log f(x) + g(x) = c $$ where $f(x)$ and $g(x)$ are rational functions, $c$ is a constant. Is there any general(in closed form) ...
2
votes
0answers
107 views

What is logarithm & how log table can be constructed by me?

I'm studying properties of logarithm but I don't understand how base e works. Base 10 looks simple while doing calculations of numbers having multiple of 10. As other numbers are not multiple of 10 ...
2
votes
0answers
230 views

Comparing Large Exponents with different bases.

How to compare large exponents with different bases? Is there any way to roughly approximate their values? For example, sort the elements of list below based on their magnitude. $381600^{809197},...
2
votes
0answers
71 views

Tweaking Reddit's Ranking Algorithm

This image explains how Reddit's Ranking algorithm works. As you know, Reddit is a very high traffic site. Therefore, the post rank decreases quite fast. This algorithm puts emphasis on bringing ...
2
votes
0answers
57 views

Solve $x=C \log(C \log(x+A)+B)$

Is it possible to resolve an equation of the type $$x=C\log{(C\log{(x+A)}+B)}$$ (where $A$, $B$, and $C$ are real-valued parameters) for $x$? As far as I can see, the function on the right hand ...
2
votes
0answers
137 views

Is this equal ? (I found it on this website)

I found this equation on this website! I would like to know it its true or not? And how can proof or disprove it?! Euler-Mascheroni constant expression, further simplification $$\ln(n)=\lim_{M\...
2
votes
0answers
29 views

Asymptotic solution to $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$

What is the smallest $t$ statisfying the inequality: $m \leqslant e^{\lambda t} (c t^q - \varepsilon)$, where $\varepsilon$ is arbitrary small positive number? I believe $t$ must be of the from: $$t = ...
2
votes
0answers
42 views

Proof of an inequality about primes

I'm very new to number theory and looking for a proof of the following inequality: $$c' \log^{\text{#} \mathbb{P}}{R} \leq \sum \limits_{\substack{n \leq R\\p|n \implies p \in \Bbb P}} 1 \leq c \log^{...
2
votes
0answers
112 views

Finding how many solutions does $f(x)=\ln x-kx$ has for $k>\frac 1 e$ and logarithmic inequality question

Find how many solutions does $f(x)=\ln x-kx$ has for $k>\frac 1 e$. $f<0$ at $x\to \infty$ and $x\to 0$. The derivative has a solution only at $x=\frac 1 k$. So place that point in $f$ and we'...
2
votes
0answers
32 views

Taking logs multiple times

Is there a formule to calculate (log (log ( log ... log n))) assume all the base to be the same (b)? I was not able to find one on wikipedia.
2
votes
0answers
94 views

Solve for $x: \ln(x+4)+\ln(x-2)=5$

Solve for x: $\ln(x+4)+\ln(x-2)=5$ Where do I go from here? If there weren't four terms in the equation I would use the quadratic formula. How can I solve for x? EDIT 1: Is this correct? ...
2
votes
0answers
55 views

Name of Logarithmic Curve

I was playing with the Desmos graphing calculator and "discovered" the following curve $(\ln x)^2 + (\ln y)^2 = 1$ (I originally found it in the parametric form $(e^{\cos t}, e^{\sin t})$). It would ...
2
votes
0answers
178 views

Comparing asymptotic growth of logarithmic functions by reasoning

As an exercise, we're sorting functions according to their asymptotical growth. When comparing these two functions, I'm getting stuck: $n^2/(\log_2 n)^3$ versus $n \log_2 n$. Using limits I am ...
2
votes
0answers
34 views

Equivalence of criteria using logarithmic transformation

Is the following criterion: $$ \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial^2 f}{\partial y\partial x} $$ Equivalent to: $$ \frac{\partial^2 \ln f}{\partial x\partial y} = \frac{\...
2
votes
0answers
53 views

Interpolation of iterated logarithms

$$\text{Let }\log^2(x)=\log(\log(x)),\\ \text{ then }f(y,x)=\log^{\lfloor1+y\rfloor}\left(\log(x)/\log((1-x^{1/x}(y-\lfloor y\rfloor))+(y-\lfloor y\rfloor))\right)$$ gives an interpolation between $\...
2
votes
0answers
42 views

Yacov Perelman Nepero game

This is my first question, so sorry if I'll make any mistake in using the site formatting. I found this game on a book by Yacov Perelman and I thought it could be nice to introduce Nepero number to ...
2
votes
0answers
88 views

Moving the branch cut of the complex logarithm

The complex logarithm is defined as $\log z:=\operatorname{Log} |z|+i\arg z$ , with the branch cut on the non-negative real axis. Determine a branch of $f(z)=\log(z^3-2)$ that is analytic at $z=0$ ...
2
votes
0answers
58 views

Integral of Difference of Logs

I get the expansion of $h$ to be $$ h(z) = {1 \over z } \sum_{r=1}^{\infty}{1 \over r}{(-{\alpha \over z}})^r $$ $$ \Rightarrow h(z) = \sum_{r=-2}^{-\infty}{{(-\alpha)^{r+1} \over -(r+1)} z^{r}} $$ ...
2
votes
0answers
42 views

When this inequality true?

If $a$ and $b$ are non-negative integers and $c$ and $d$ are non-negative real numbers, for what values is the following inequality true? $\log((a+b)!) - \log(a!b!) \ge(a+b) \log(c+d) - (a \log(c) +...
2
votes
0answers
98 views

Showing that a logarithmic inequality holds

Given $0 < x_1 < x_2 < x_3 < x_4 < 1$, how can I show that the following inequality holds: $$ \frac{1}{R(x_1, x_3)}+\frac{1}{R(x_2, x_4)}<\frac{1}{R(x_1, x_2)}+\frac{1}{R(x_3, x_4)} ...
2
votes
0answers
56 views

show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$, and $0<a<b$

Show that $(x^b+y^b)^{1/b} < (x^a+y^a)^{1/a}$ if $x>0$, $y>0$,and $0<a<b$ by examining the sign of the derivative of an appropriate function. This is an exercise in middle part of ...