Questions related to real and complex logarithms.

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1answer
57 views

Looking for help understanding the asymptotic expansion of the digamma function

I was recently given an example using this asymptotic expansion of the digamma function where: $$\frac{d}{dx}(\ln\Gamma(x)) = \psi(x) \sim \ln(x) - \frac{1}{2x} - \frac{1}{12x^2}$$ Here's the ...
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4answers
210 views

To find the logarithm of $1728$ to the base $2 \sqrt{3}$

Find the logarithm of: $1728$ to base $2\sqrt{3}$. Let, $\log_{2\sqrt{3}} 1728 = y$, then $$\begin{align} (2\sqrt{3})^y &= 1728\\ 2^y(\sqrt3)^y &= 1728\\2^y(3^\frac12)^y &= ...
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1answer
61 views

Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} < 1$?

This is the second attempt at a proof. My first attempt had a flaw in its logic. After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved. The revision ...
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0answers
45 views

Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} \ge 1$?

This is the second attempt at a proof. My first attempt had a flaw in its logic. After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved. The revision ...
4
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1answer
121 views

Need help understanding if a function is increasing or decreasing

I am working on understanding the following function: $$g(x) = \ln\Gamma\left(\frac{x}{4}\right) - \ln\Gamma\left(\frac{x}{5}+\frac{1}{2}\right) - \ln\Gamma\left(\frac{x}{20}+\frac{1}{2}\right) - ...
8
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1answer
245 views

Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?

I am starting to feel more confident in my understanding of Ramanujan's proof of Bertrand's postulate. I hope that I am not getting overconfident. In particular, Ramanujan's does the following ...
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4answers
3k views

Why aren't logarithms defined for negative $x$?

Given a logarithm is true, if and only if, $y = \log_b{x}$ and $b^y = x$ (and $x$ and $b$ are positive, and $b$ is not equal to $1$)[1], are true, why aren't logarithms defined for negative ...
2
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1answer
101 views

The rate of increase of the Gamma Function over real numbers

If $$ x_1 > x_2 > 0$$ and $$\Delta{x}>0$$ does it follow that: $$\ln\Gamma(x_1 + \Delta{x}) - \ln\Gamma(x_1) \ge \ln\Gamma(x_2 + \Delta{x}) - \ln\Gamma(x_2)$$ Would it be enough to show ...
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4answers
169 views

How do i find the inverse of: $f(x) = {2^{x - 1}} - 3$

$f(x) = 2^{x - 1} - 3$ My approach: Take logs to base 2: $ = \log_2 \left( x - 1 \right) - \log_2 \left( 2^3 \right)$ $ = \log_2 \left( {x - 1} \over {2^3} \right)$ This isn't the answer in the ...
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2answers
68 views

Prove that $\ln$ has an inverse function

For $x$ in $(0, \infty)$ let $\ln(x) = \int_{1}^{x}\frac{1}{t}dt$. Prove that $\ln$ has an inverse function My book does not really go into how to prove something has an inverse, besides it needing ...
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0answers
144 views

Using the gamma function as an upper and lower bound to the logarithm of a factorial function.

I am trying to find an upper and lower bound for the following function: $$f(x) = \ln(\lfloor\frac{x}{b_1}\rfloor!) - \ln(\lfloor\frac{x}{b_2}\rfloor!) - \ln(\lfloor\frac{x}{b_3}\rfloor!)$$ where ...
18
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5answers
2k views

Evaluating $\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx$

How would I go about evaluating this integral? $$\int_0^{\infty}\frac{\ln(x^2+1)}{x^2+1}dx.$$ What I've tried so far: I tried a semicircular integral in the positive imaginary part of the complex ...
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3answers
300 views

Why I can't calculate $0*log(0)$ but can $log(0^0)$

I got this doubt after some difficult in programming. In a part of code, i had to calculate: $$ x = 0 * Log(0) \\ x = 0*-Inf $$ and got $x = NaN$ (in R and Matlab). So I changed my computations to ...
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1answer
252 views

Understanding the upper and lower bounds of the error estimate in Stirling's Approximation

Based on the Wikipedia article on Stirling Approximation: $n! = \sqrt{2 \pi n} \left(\frac{n}{e}\right)^n e^{\lambda_n}$ where $\frac{1}{12n+1} < \lambda_n < \frac{1}{12n}$ How would this ...
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1answer
1k views

Generalizing Ramanujan's proof of Bertrand's Postulate: Can Ramanujan's approach be used to show a prime between $4x$ and $5x$ for $x \ge 3$

Perhaps, I've been thinking too long about Ramanujan's proof, but it appears to me that his argument can be generalized beyond $x$ and $2x$. My argument below attempts to show that for $x \ge 1331$, ...
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1answer
37 views

Some number to an exponent of a log

I need to simplify an expression. I am currently working on the following problem (I apologize in advance for formatting, I'm not sure how to use it on Stack Exchange): $81^{(\log_{3}N)+(log_{9}N)}$ ...
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1answer
65 views

How to compute a product of logarithms?

I've been reading through Stewart's Calculus textbook, and came across the following problem fairly early on - What is $$\prod_{i = 2}^{31} \log_i (i + 1)\;?$$ I did some searching, and found ...
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1answer
199 views

Discrete logarithm - strange polynomials

If $p$ is a prime number and $\omega$ is a fixed primitve root for $\mathbb{Z}/p\mathbb{Z}$, then we can define the discrete logarithm of $x \in (\mathbb{Z}/p\mathbb{Z})^{\times}$ as the unique number ...
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1answer
41 views

Sufficient conditions for an inequality with a log

I need to find sufficient conditions so that $x \geq \frac{1}{a-\ln{x}}$ for $a>1$ and $x > 0$. Is there a way to get a tight solution to the problem?
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2answers
67 views

Logarithm calculation result

I am carrying out a review of a network protocol, and the author has provided a function to calculate the average steps a message needs to take to traverse a network. It is written as ...
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2answers
265 views

Analyzing the lower bound of a logarithm of factorials using Stirling's Approximation

I am trying to get the lower bound for: $f(x) = \ln(\lfloor\frac{x}{4}\rfloor!) - \ln(\lfloor\frac{x}{5}\rfloor!) -\ln(\lfloor\frac{x}{20}\rfloor!) - 2(1.03883)(\sqrt{\frac{x}{4}}) - ...
2
votes
3answers
225 views

Showing $\log(2)$ and $\log(5)$

How do I show that: $$\log(2)=\sum^\infty_{n=1}(-1)^{n+1}\frac{1}{n}$$ and that $$\log(5)=\log(3)+\sum^\infty_{n=1}(-1)^{n+1}\frac{2^n}{n3^n}$$ Thanks in advanced.
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5answers
106 views

Getting stuck on simple logarithmic equation

$$x \times \ln (x) = 1$$ I am trying to solve that equation. I used the theory $ln(a) = ln(b)$ being equivalent to $a = b$ and got stuck at $$x = e^{\frac{1}{x}}$$ That's as far as I went and I ...
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1answer
841 views

Function design: a logarithm asymptotic to one?

I want to design an increasing monotonic function asymptotic to $1$ when $x\to +\infty $ that uses a logarithm. Also, the function should have "similar properties" to $\dfrac{x}{1+x}$, i.e.: ...
19
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5answers
2k views

Evaluate: $\int_0^{\pi} \ln \left( \sin \theta \right) d\theta$

Evaluate: $ \displaystyle \int_0^{\pi} \ln \left( \sin \theta \right) d\theta$ using Gauss Mean Value theorem. Given hint: consider $f(z) = \ln ( 1 +z)$. EDIT:: I know how to evaluate it, but I am ...
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1answer
45 views

Is it true that $\int t\frac{dF}{d \ln{t}} d \ln{t}=\int \frac{dF}{dt} dt$

It seems to be true that: $$\int t\frac{dF}{d \ln{t}} d \ln{t}=\int \frac{dF}{dt} dt$$ For eg., this works with $\frac{dF}{dt}=\frac{1}{2} (\cos(\pi \ln{t})+1)$ But then there must be something ...
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3answers
1k views

Finding the derivative of a function with a Natural Log.

I am trying to differentiate the function: $${\rm ln} \left(\frac{3x \ {\rm tan}(x)}{x^2 + 2}\right)$$ I think step one is to use the quotient rule of natural log expanding the expression. However ...
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1answer
133 views

Derivatives with Natural Log (Help)

This is the problem: $$f(x)=\ln[\sin(-2x)\cos(-2x)]$$ This is as far as I can get: $$\frac{-2[\cos(-2x)]}{\sin(-2x)}+\frac{2[\sin(-2x)]}{\cos(-2x)}$$ I'm familiar with the rules of differentiation ...
4
votes
2answers
410 views

Double integral application

I need to determine $$\int_{0}^{1} \int_{-\sqrt{x}}^{\sqrt{x}}\frac{1}{1-y}dydx$$ I integrate in terms of the y component and obtained: $$\int_{0}^{1}\ln(\frac{1+\sqrt{x}}{1-\sqrt{x}})dx$$ Can ...
3
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1answer
250 views

Reasoning about the Chebyshev functions: How does one check an upper bound based on the second Chebyshev function?

In Ramanujan's proof of Bertrand's Postulate, Ramanujan states: $\log([x]!) - 2\log([\frac{1}{2}x]!) \le \psi(x) - \psi(\frac{1}{2}x) + \psi(\frac{1}{3}x)$ where: $\vartheta(x) = \sum_{p \le x} ...
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0answers
506 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 ...
3
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2answers
133 views

Solve $-B \ln y -A y \ln y + A y- A =0$ for $y$

I would like to know if there is a (preferably closed-form) solution for $-B \ln y -A y \ln y + A y- A =0$ for $y$ Where $A, B \in \mathbb{R}^{+}$. I have reasons to think there isn't a closed form ...
1
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1answer
40 views

What is a $\log_{10} \%$ transfer?

I have this graph of results comparing the transfer percentages of bacteria to hands with and without gloves. By the looks of things, the higher the bacteria count on the chicken the lower the ...
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votes
2answers
138 views

Solve $B \ln y +A y \ln y + A y-A =0$ for $y$

I would like to know if there is a (preferably closed-form) solution for $B \ln y +A y \ln y + A y- A =0$ for $y$ Where $A, B \in \mathbb{R}^{+}$. I have reasons to think there isn't a closed form ...
0
votes
1answer
40 views

Exponent, logarithmic question

I'm reading an article related to bioinformatics and I found this formula: Probability of $x =(1-y/n)^t$ or approximately $e^{-yt/n}$. My question is how do we pass to the approximation given in the ...
3
votes
2answers
61 views

Separating $\frac{1}{1-x^2}$ into multiple terms

I'm working through an example that contains the following steps: $$\int\frac{1}{1-x^2}dx$$ $$=\frac{1}{2}\int\frac{1}{1+x} - \frac{1}{1-x}dx$$ $$\ldots$$ $$=\frac{1}{2}\ln{\frac{1+x}{1-x}}$$ I ...
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4answers
217 views

How can i simplify $b^\frac{\ln a}{\ln b}$?

What rules can i use to simplify $b^\frac{\ln a}{\ln b}$ for $a,b>1$ ?
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1answer
148 views

How can I simplify this logarithmic expression

$lg\lceil \frac{n}{2} \rceil + 1$ How do I get rid of the ceiling? In order to lose the ceiling I add +1 and get the following expression which I don't know how to simplify $lg (\frac{n +1} {2}) + ...
2
votes
1answer
4k views

how to expand the following -> $\log (x + y)$

I know for a fact that it is not $\log x + \log y$, but Im unsure as to how to proceed.. I have checked the basic log properties but nowhere do they give an example of a statement like the one above. ...
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3answers
429 views

Exponential and logarithmic series: Find the sum of $2^2 + 3^2/ 2!+4^2/3!+…$ to infinity

Find the sum of the following series: $ 2^2 + 3^2/2! + 4^2/3! + ...$ to infinity The answer is given as $5e$ but I got it as $5e+1$ $T_n = 1/(n-2)! +3/(n-1)! + 1/(n)! $ for $n \ge 2$ and ...
2
votes
1answer
86 views

Finding the min and max of $f(x) = \log_{10}x + x^3 - x^2 - 6x + 3$

$$f(x) = \log_{10}x + x^3 - x^2 - 6x + 3$$ $$x > 0$$ How do I find the maxima and the minima of this function? This is a highscool level problem.
2
votes
1answer
159 views

why $\log(n!)$ isn't zero?

I have wondered that why the $\log (n!)$ isn't zero for $n \in N$. Because I think that $\log (1)$ is zero so all following numbers after multiplying the result will become zero. Thanks in ...
2
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4answers
99 views

Simplify a logarithm function

$$2\log\sqrt[4]{10}-\ln e^{-7}+\log_9\sqrt 3$$ I want to simplify this function. I believe that $\,2\log\sqrt[4]{10}\,$ can become $\,\log\sqrt{10}\,$ but now I'm stuck. Is it possible that $\ln ...
2
votes
2answers
70 views

logarithm problem - four tuple

How many distinct four tuple (a,b,c,d) of rational numbers are there with $a\log_{10}2+b\log_{10}3+c\log_{10}5+d\log_{10}7=2005$ Can we proceed like this : Using $\log a +\log b = \log(ab)$ and ...
3
votes
3answers
241 views

Mathematical Statistics

How do I find the answers to this question? State Tech’s basketball team, the Fighting Loga- rithms, have a 70% foul-shooting percentage. (a) Write a formula for the exact probability that out of ...
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2answers
78 views

$ 10^{-9}[2\times10^6 + 3^{1000}] $

$$ 10^{-9}[2\times10^6 + 3^{1000}] $$ I'm stuck on solving this. I wasn't able to put this into my calculator since the number is too big for it to calculate. So far I've done this: $$ ...
1
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2answers
2k views

Limit, log rule

I have couple of question on this part of equation - \begin{align} \lim_{n\to \infty } \frac{ 7 \cdot \sqrt{n}}{\log(n)}- \lim_{n\to \infty} \frac{1}{n\cdot \log(n)} &=\lim_{n\to \infty} \frac{7 ...
6
votes
2answers
206 views

Derivative of ${ x }^{ x }$ without logarithmic differentiation

With logarithmic differentiation, it is quite simple to compute the derivative of $x^x$: $$y=x^x$$ $$\ln {y} =x \ln{x}$$ $$\frac {1}{y} \frac {dy}{dx} = \ln{x} +1$$ $$\frac {dy}{dx} ...
1
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2answers
69 views

Finding the coefficient of $ x^n $ in the expansion of $ { ({\ log_e (1+x) })^2 } $

I've been trying to find the coefficient of $x^n$ in the expansion of $ { ({\log_e (1+x) })^2 } $.I wrote out the expansion of $ { ({\log_e (1+x) })^2 } $ explicitly and tried to generalize the terms ...
0
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3answers
124 views

Question about changing a logarithm's base

I've been using the following method to derive/remember the logarithm base conversion formula: If I want to convert $\log_a(x)$ to an expression in base $b$, I say, $$a^{\log_a(x)}=x\\ ...