Questions tagged [logarithms]

Questions related to real and complex logarithms.

984 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
18
votes
0answers
505 views

Prove a strong inequality $\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2-\frac{7\ln 2}{8\ln n}\right)\sum_{k=1}^n\frac 1{a_k}$

For $a_i>0$ ($i=1,2,\dots,n$), $n\ge 3$, prove that $$\sum_{k=1}^n\frac{k}{a_1+a_2+\cdots+a_k}\le\left(2\color{red}{-\frac{7\ln 2}{8\ln n}}\right)\sum_{k=1}^n\frac 1{a_k}.$$ The case without $\...
15
votes
0answers
908 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 ...
10
votes
0answers
256 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 ...
9
votes
0answers
422 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 ...
6
votes
1answer
103 views

Logarithm simultaneous equation

Let $(x_1, y_1,z_1)$ and $(x_2, y_2, z_2)$ - where $x_1\ge y_1\ge z_1$ and $x_2\le y_2\le z_2$ - be two triplets satisfying the following simultaneous equations: $$ \begin{align} \log_{10}(2xy)&...
6
votes
0answers
113 views

Solutions to $x \exp(x) = a x + b$

Are there any closed form solutions to $$ x \exp(x) = a x + b $$ for real-valued $x$, $a$, and $b$ (we wish to solve for $x$, and $a$ and $b$ are simple constants)? We can assume that everything is ...
6
votes
0answers
179 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
110 views

Evaluate $\int_0^1 \log^n(x^a)\log^m(1-x^{\color{red}{\alpha}})x^b(1-x^{\color{red}{\beta}})^t\mathrm dx$ with $\alpha\ne\beta$

Recently dealing with algebraic integrals composited of logarithms and polynomials. I learned about using the derivatives of the Beta Function in order to evaluate them. Applying this knowledge I was ...
5
votes
0answers
82 views

Inequality involving power to fractional part of integer multiples of logarithm of integer to coprime base.

For $x \in \mathbb{R}^+$, let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$. Let $k \in \mathbb{N}$. Show that $2^{\{k \log_2(3)\}} < \dfrac{2}{1 + 2^{-k}}$ for $k > 1$. ...
5
votes
0answers
133 views

Nasty logarithmic infinite sums arising from a tan integral

While trying to work on an old post, I managed after some manipulations to show that the nasty improper tan integral $$ I=\int_0^{\pi} \left( \frac{\pi}{2} - x \right) \frac{\tan x}{x} \, {\rm d}x\...
5
votes
0answers
79 views

Computing: $\int_0^{\infty}\frac{dx}{(x+a)((\ln x)^2+\pi^2)}.$

Assume $a>0$ and evaluate $$\int_0^{\infty}\frac{dx}{(x+a)((\ln x)^2+\pi^2)}.$$ My biggest issue when doing these problems is deciding which closed curve I should try, so if someone could point me ...
5
votes
0answers
63 views

There exists infinitely many $N$ such that $\{\sum_{n=2}^N\log(n)\}<\varepsilon$

I am wondering whether or not the following result is true: For all $\varepsilon>0$, there exists infinitely many $N\in \mathbb N$ such that $$S_N:=\left\{\sum_{n=2}^N\log(n)\right\}<\...
5
votes
1answer
168 views

Derivative of $f(x)^{g(x)}$ at points when $f(x)=0$

I am interested in understanding the general behavior of the derivative for $$f(x)^{g(x)}$$ at points where $f(x)=0$. For example, if $f^g=x^n$ we have $$\frac{d}{dx}f^g(0)=\begin{cases}0 & n\ge ...
5
votes
1answer
251 views

Does a logarithmic branch point imply logarithmic behavior?

The complex logarithm $L(z)$ is given by $$L(z)=\ln(r)+i\theta$$ where $z=re^{i\theta}$ and $\ln(x)$ is the real natural logarithm. It is well known that $L(z)$ then sends each $z$ to infinitely many ...
5
votes
0answers
2k 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
2answers
162 views

Is there a clean way to show $\frac{\log(x-\tfrac12\log x)}{\log(x+\tfrac12\log x)}>1-\tfrac1x$ for all $x>1$?

In writing a paper I had to show that the inequality holds for large enough $x$, which is easy, but I ended up being pretty sure it holds for all $x>1$, so I would like to include the proof of the ...
4
votes
0answers
144 views

Closed form for $\ln(2\ln(3\ln(4\ln(5\ln(6…)))))$

A recent question asked whether the infinite nested logarithm $\ln(2\ln(3\ln(4\ln(5\ln(6…)))))$ has a finite value. It does: I showed with a crude method that the value was at most $2$, and Hagen von ...
4
votes
0answers
93 views

Can we write $\ln(x) $ as an infinite sum of $n$ th roots?

Is there a real sequence $0 \leq a_n $ such that for $x > 1 $ we have : $$\ln(x) = a_0 + \sum_{n=1}^{\infty} (-1)^n \cdot a_n \cdot x^{1/n}$$ Or $$\ln(x) = a_0 + \sum_{n=1}^{\infty} (-1)^{1+n} ...
4
votes
0answers
214 views

Evaluate $\int_0^1 \frac{\log(1-z)\log(1-z^3)}{z^2}dz$

Integrals of the kind $$\int_0^1 \frac{\log(1-z)\log(1-z^n)}{z^2}dz$$ where $n\geq 1$ is an integer, arises from a natural way when one apply Möbius inversion to get identities realated to $\zeta(2)$ (...
4
votes
0answers
105 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, $$ f(xy) = f(x) + f(y), $$ ...
4
votes
1answer
313 views

Name for a Logarithm Identity/Property

I came across a neat logarithm fact today: $\large n^{\log_bx} = x^{\log_bn}$ One simple proof is: $\large \log_bx\cdot \log_bn=\log_bx\cdot \log_bn$ $\large \Rightarrow \log_bx^{log_bn}=...
4
votes
2answers
132 views

Limit of $x^x$ as $x$ tends to $0$

I am trying to solve the following limit: $$\lim \limits_{x\to0} x^x$$ The only thing that comes to mind is to write $x^x$ as $e^{x\ln{x}}$ and getting the right sided limit would be easy but I don'...
4
votes
0answers
137 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
139 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
1answer
353 views

Series involving Logs

I'm trying to find the name of, and a good online reference to, a type of "logarithm series", e.g. $$(1+x)^9 = \sum_{k=0}^{\infty} \frac{9^k\ln^k(1+x)}{k!} $$ I realise that this comes from $x^y \...
3
votes
0answers
45 views

Generalizing Oksana's trilogarithm relation to $\text{Li}_3(\frac{n}7)$?

This was inspired by Oksana's post. Let $$a = \ln 2\\ b = \ln 3\\ c = \ln 5$$ Then the following sums can be expressed in terms of $a,b,c,$ $$A = \text{Li}_3\left(\frac12\right)\\ B = \text{Li}_3\...
3
votes
0answers
34 views

Exponential touch Logarithm

When I write graph for fun I found that there is some value a that make $y=a^x$ and $y=\log_a (x)$ don't touch each other and there is some value a that make $y=a^x$ and $y=\log_a (x)$ intersect each ...
3
votes
0answers
75 views

“Straightforward” estimate on intersection-probabilities of Brownian Motion

I am currently working on a paper and there appears this so called "straightforward estimate": $$\mathbf{P}\{B[0,1] \cap B[3,n] = \emptyset\} \leq \frac{c}{\ln(n)} \quad \text{where}\ c<\infty\ \...
3
votes
1answer
77 views

Is it known that $\sum_{i=1}^\infty \frac{if\;(i \pmod n=0)\;then\;(1-n)\;else\;(1)}{i}=log(n)$?

I found this general infinite sum: $\sum_{i=1}^\infty \frac{ \mathtt i \mathtt f \; (i \pmod n = 0) \; \mathtt t \mathtt h \mathtt e \mathtt n \;(1-n) \; \mathtt e \mathtt l \mathtt s \mathtt e \; (...
3
votes
0answers
66 views

Closed form of Integral of ellipticK and log using Mellin transform? $\int_{0}^4 K(1-u^2) \log[1+u z] \frac{du}{u}$

I am trying to evaluate the integral: $\mathcal{I}(z)=\int_{0}^a K(1-u^2) \log[1+u z] \frac{du}{u}$, $\qquad $ ($a$ fixed, $a>0$ and $K$ is the complete elliptic integral of the first kind) in ...
3
votes
0answers
37 views

Pushing the Iterated Logarithm Rule to the limits.

Given a standard Brownian motion, the Iterated Logarithm Rule says that with probability one, $$\frac{|w(t)|}{\sqrt{t \log\log t}},\ (1\le t)$$ has $\limsup$ $\sqrt{2}$ as $t \to\infty$. But what is ...
3
votes
0answers
39 views

A function that turns a power to a product/sum?

I just recently started learning about the logarithm functions, and its concept is quite amazing (f(x.y)=f(x)+f(y)). Now I'm asking for a similar function but instead of x.y ; we use ...
3
votes
0answers
38 views

compute $\log_e(j)$ of split complex number $j$

I am trying to calculate the value of $\ln j$ where $j^2=1, j\ne\pm1$($j$ is split complex). this is how i did it: given $e^{j\theta}=\cosh\theta+j\sinh\theta$ I can set $\cosh\theta=0\implies \...
3
votes
1answer
39 views

Asymptotic notation question (logarithms)

I am working through some problems and I have come across one I do not understand. Could someone clarify why $$2x^3 + 3x^2\log(x) + 7x + 1$$ is $O(x^3\log(x))$ for $x>0$? I guess I am missing ...
3
votes
0answers
145 views

branch cut for complex logarithm involving composition with a square root

Here is the problem I am working on: PROBLEM: Let $f(z) = \log(z+(z^2-1)^{1/2})$. The branch cut for $(z^2 - 1)^{1/2}$ is to be the portion of the real axis $[-1,1]$. (a) Show that $(z^2 - 1)^{1/2}$...
3
votes
0answers
70 views

Limit involving a square root and an inverse Laplace transform

How can I find a general form of the equation: $$\lim_{x\to\infty}\sqrt{\frac{1}{x}\int_0^x\left(\mathscr{L}_\text{s}^{-1}\left[\text{F}\left(\text{s}\right)\cdot\text{G}\left(\text{s}\right)\...
3
votes
0answers
220 views

Interesting integral involving Laplace transform and the sine of ln

Question: Find a closed form for $\mathscr{I}(\text{n})$: $$\mathscr{I}(\text{n}):=\int_0^\infty\frac{\sin(\ln(x))\cdot\cos\left(\frac{\text{n}} x\right)}{x}\space\text{d}x\tag1$$ Using ...
3
votes
2answers
113 views

Why is the plot of some functions so similar to the plot of ln(x)?

Using https://www.desmos.com/calculator and my calculus knowledge (the integral power rule $\int x^n dx= (x^n+1)/(n+1)+C$ and the exception $\int x^{-1}dx=ln(x)+C$), I have noticed that functions like ...
3
votes
0answers
55 views

Logarithm properties in fields in general

I know that the logarithm function can be defined in a field, as an inverse of the exponential function (from here), if the exponential function is defined such that it is bijective, but I'd like to ...
3
votes
0answers
52 views

Are $\{f>0 ~ \& ~ \frac{\partial^3 f}{\partial x^3} ~ \text{exists}\}$ and $\{\frac{\partial^3 \log f}{\partial x^3} ~ \text{exists}\}$ equivalent?

Question. Are the following conditions equivalent? \begin{align} \textbf{Condition 1}: \quad & f > 0 ~ \text{and} ~ \frac{\partial^{3} f}{\partial x^{3}} ~ \text{exists}. \\ \textbf{Condition ...
3
votes
0answers
131 views

Connecting $\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$ and $\sum\limits_{n=1}^\infty\frac{1}{n \binom{3n}{n}}$

Define three generalized hypergeometric functions (which differs in the starting numerator in blue): $$A_k =\,_kF_{k-1}\left(\left.\begin{array}{c} 1,\frac{\color{blue}{k-2}}{k-1},\frac{k-1}{k-1},\...
3
votes
0answers
580 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 < \...
3
votes
1answer
114 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
66 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
288 views

Prob. 7, Chap. 1 in Baby Rudin

Here's problem 7 in the exercises following Chap. 1 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Fix $b > 1$, $y > 0$, and prove that there is a unique real number $x$ ...
3
votes
0answers
16 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
74 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
1answer
57 views

Deriving a formula

I'm not sure whether I should post this on the chemistry stack exchange or the mathematics since it mainly consists of mathematical knowledge but also has some chemistry terms in it. Note: I am not ...
3
votes
0answers
918 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}, ...
3
votes
1answer
86 views

Solving equations with logarithmic exponent

I need to solve the equation : $\ln(x+2)+\ln(5)=\lg(2x+8)$ With the change of base formula we can turn this into: $\ln(x+2)+\ln(5)=\frac{\ln(2x+8)}{\ln(10)}$ We can also simplify the LHS with the ...