# Questions tagged [logarithms]

Questions related to real and complex logarithms.

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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 $\... 0answers 908 views ### Understanding Ramanujan's approach in his proof of Bertrand's Postulate I've been reading through Ramanujan's proof of Betrand's Postulate and I'm not clear why he didn't state his proof in terms of$\varphi(2x) - \varphi(x)$What would be wrong with this approach for ... 0answers 256 views ### The diferential equation$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 ... 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 ... 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)&... 0answers 113 views ### Solutions to x \exp(x) = a x + b Are there any closed form solutions to x \exp(x) = a x + b $$for real-valued x, a, and b (we wish to solve for x, and a and b are simple constants)? We can assume that everything is ... 0answers 179 views ### An integral to prove that \log(2n+1) \ge H_n Dalzell integral The equation$$ \int_0^1 \frac{x^4(1-x)^4}{1+x^2}dx=\frac{22}{7}-\pi $$proves that \frac{22}{7}-\pi>0 because the integrand is positive. Some Dalzell-type integrals for \log(... 0answers 110 views ### Evaluate \int_0^1 \log^n(x^a)\log^m(1-x^{\color{red}{\alpha}})x^b(1-x^{\color{red}{\beta}})^t\mathrm dx with \alpha\ne\beta Recently dealing with algebraic integrals composited of logarithms and polynomials. I learned about using the derivatives of the Beta Function in order to evaluate them. Applying this knowledge I was ... 0answers 82 views ### Inequality involving power to fractional part of integer multiples of logarithm of integer to coprime base. For x \in \mathbb{R}^+, let \{x\} = x - \lfloor x \rfloor denote the fractional part of x. Let k \in \mathbb{N}. Show that 2^{\{k \log_2(3)\}} < \dfrac{2}{1 + 2^{-k}} for k > 1. ... 0answers 133 views ### Nasty logarithmic infinite sums arising from a tan integral While trying to work on an old post, I managed after some manipulations to show that the nasty improper tan integral$$ I=\int_0^{\pi} \left( \frac{\pi}{2} - x \right) \frac{\tan x}{x} \, {\rm d}x\... 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 ... 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 $$S_N:=\left\{\sum_{n=2}^N\log(n)\right\}<\... 1answer 168 views ### Derivative of f(x)^{g(x)} at points when f(x)=0 I am interested in understanding the general behavior of the derivative for$$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 ... 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 ... 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 ... 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 ... 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 ... 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 $$\ln(x) = a_0 + \sum_{n=1}^{\infty} (-1)^{1+n} ... 0answers 214 views ### Evaluate \int_0^1 \frac{\log(1-z)\log(1-z^3)}{z^2}dz Integrals of the kind$$\int_0^1 \frac{\log(1-z)\log(1-z^n)}{z^2}dz$$where n\geq 1 is an integer, arises from a natural way when one apply Möbius inversion to get identities realated to \zeta(2) (... 0answers 105 views ### Is the product rule for logarithms an if-and-only-if statement? If a function f(x) is proportional to \ln x, then we know$$ 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), $$... 1answer 313 views ### Name for a Logarithm Identity/Property I came across a neat logarithm fact today: \large n^{\log_bx} = x^{\log_bn} One simple proof is: \large \log_bx\cdot \log_bn=\log_bx\cdot \log_bn \large \Rightarrow \log_bx^{log_bn}=... 2answers 132 views ### Limit of x^x as x tends to 0 I am trying to solve the following limit:$$\lim \limits_{x\to0} x^x$$The only thing that comes to mind is to write x^x as e^{x\ln{x}} and getting the right sided limit would be easy but I don'... 0answers 137 views ### Trying to show that \ln(x!) - \ln(\lfloor\frac{x}{2}\rfloor!) - \ln(\lfloor\frac{x}{3}\rfloor!) - \ln(\lfloor\frac{x}{6}\rfloor!) \ge \psi(x) I've been told that the approach below will not work. I would be interested if someone could help me to understand what will go wrong. Let:$$\psi(x) = \sum\limits_{p^k \le x} \ln p$$So that (see ... 0answers 139 views ### Generalizing Jitsuro Nagura's result: my resulting upper bound for the second chebyshev function is too low. What am I doing wrong? I've been reading through Jitsuro Nagura's classic proof that there is a prime between x and \frac{6x}{5} and it seems to me that it should be possible to improve on his upper bound for the second ... 1answer 353 views ### Series involving Logs I'm trying to find the name of, and a good online reference to, a type of "logarithm series", e.g.$$(1+x)^9 = \sum_{k=0}^{\infty} \frac{9^k\ln^k(1+x)}{k!} $$I realise that this comes from x^y \... 0answers 45 views ### Generalizing Oksana's trilogarithm relation to \text{Li}_3(\frac{n}7)? This was inspired by Oksana's post. Let$$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\... 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 ... 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": $$\mathbf{P}\{B[0,1] \cap B[3,n] = \emptyset\} \leq \frac{c}{\ln(n)} \quad \text{where}\ c<\infty\ \... 1answer 77 views ### Is it known that \sum_{i=1}^\infty \frac{if\;(i \pmod n=0)\;then\;(1-n)\;else\;(1)}{i}=log(n)? I found this general infinite sum: \sum_{i=1}^\infty \frac{ \mathtt i \mathtt f \; (i \pmod n = 0) \; \mathtt t \mathtt h \mathtt e \mathtt n \;(1-n) \; \mathtt e \mathtt l \mathtt s \mathtt e \; (... 0answers 66 views ### Closed form of Integral of ellipticK and log using Mellin transform? \int_{0}^4 K(1-u^2) \log[1+u z] \frac{du}{u} I am trying to evaluate the integral: \mathcal{I}(z)=\int_{0}^a K(1-u^2) \log[1+u z] \frac{du}{u}, \qquad (a fixed, a>0 and K is the complete elliptic integral of the first kind) in ... 0answers 37 views ### Pushing the Iterated Logarithm Rule to the limits. Given a standard Brownian motion, the Iterated Logarithm Rule says that with probability one,$$\frac{|w(t)|}{\sqrt{t \log\log t}},\ (1\le t)$$has \limsup \sqrt{2} as t \to\infty. But what is ... 0answers 39 views ### A function that turns a power to a product/sum? I just recently started learning about the logarithm functions, and its concept is quite amazing (f(x.y)=f(x)+f(y)). Now I'm asking for a similar function but instead of x.y ; we use ... 0answers 38 views ### compute \log_e(j) of split complex number j I am trying to calculate the value of \ln j where j^2=1, j\ne\pm1(j is split complex). this is how i did it: given e^{j\theta}=\cosh\theta+j\sinh\theta I can set \cosh\theta=0\implies \... 1answer 39 views ### Asymptotic notation question (logarithms) I am working through some problems and I have come across one I do not understand. Could someone clarify why$$2x^3 + 3x^2\log(x) + 7x + 1$$is O(x^3\log(x)) for x>0? I guess I am missing ... 0answers 145 views ### branch cut for complex logarithm involving composition with a square root Here is the problem I am working on: PROBLEM: Let f(z) = \log(z+(z^2-1)^{1/2}). The branch cut for (z^2 - 1)^{1/2} is to be the portion of the real axis [-1,1]. (a) Show that (z^2 - 1)^{1/2}... 0answers 70 views ### Limit involving a square root and an inverse Laplace transform How can I find a general form of the equation:$$\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)\... 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 ... 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 ... 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 ... 0answers 52 views ### Are$\{f>0 ~ \& ~ \frac{\partial^3 f}{\partial x^3} ~ \text{exists}\}$and$\{\frac{\partial^3 \log f}{\partial x^3} ~ \text{exists}\}equivalent? Question. Are the following conditions equivalent? \begin{align} \textbf{Condition 1}: \quad & f > 0 ~ \text{and} ~ \frac{\partial^{3} f}{\partial x^{3}} ~ \text{exists}. \\ \textbf{Condition ... 0answers 131 views ### Connecting\sum\limits_{n=0}^\infty\frac{n!\,(2n)!}{(3n+2)!}$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): $$A_k =\,_kF_{k-1}\left(\left.\begin{array}{c} 1,\frac{\color{blue}{k-2}}{k-1},\frac{k-1}{k-1},\... 0answers 580 views ### Why is \log z = \ln r + i\theta (r>0, \alpha <\theta < \alpha + 2\pi) discontinuous at \alpha? In one book on complex variables it is written that, given the function \log z = \ln r + i\theta (for proper citation, let's call it function (2), as in the book) (r>0, \alpha <\theta < \... 1answer 114 views ### All solutions of z^i = i^z In the simple equation z^i = i^z how are all complex values found? z= \pm \, i, and what else? It can be found by inspection, but to find general solution: We take logs, there is a ... 0answers 66 views ### Want to know what's wrong? I take a exercise from apostol's book. I was trying the next exercise and do it, but the answer (from the book) is different, and I don't know what part of my procedure it's wrong?. So I want to know ... 0answers 288 views ### Prob. 7, Chap. 1 in Baby Rudin Here's problem 7 in the exercises following Chap. 1 in Principles of Mathematical Analysis by Walter Rudin, 3rd edition: Fix b > 1, y > 0, and prove that there is a unique real number x ... 0answers 16 views ### optimal hill “rank” cannot be solved? Okay so I was thinking about the following problem today: We have a guy who is h tall stand upon a paraboloid shaped hill of the form z=-ar^n How far away (in r) does his friend who is also h ... 0answers 74 views ### Checking whether sequence x_n = \ln(n^2 + 1) - \ln(n) converges or diverges I have to show whether$$ 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).$$... 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 ... 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}, ...
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 ...