Questions related to real and complex logarithms.
1
vote
1answer
27 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 ...
1
vote
0answers
101 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
...
4
votes
1answer
109 views
Need help understanding if a function is increasing or decreasing
I am working on understanding the following function:
$$g(x) = \ln\Gamma\left(\frac{x}{4}\right) - \ln\Gamma\left(\frac{x}{5}+\frac{1}{2}\right) - \ln\Gamma\left(\frac{x}{20}+\frac{1}{2}\right) - ...
0
votes
1answer
37 views
Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} < 1$?
This is the second attempt at a proof. My first attempt had a flaw in its logic.
After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved.
The revision ...
1
vote
0answers
29 views
Is it possible to generalize Ramanujan's lower bound for factorials when $\{\frac{x}{b_2}\} + \{\frac{x}{b_3}\} \ge 1$?
This is the second attempt at a proof. My first attempt had a flaw in its logic.
After reviewing the mistake in logic, I believe that with a revised logic, the argument can be saved.
The revision ...
4
votes
3answers
101 views
To find the logarithm of $1728$ to the base $2 \sqrt{3}$
Find the logarithm of: $1728$ to base $2\sqrt{3}$.
Let, $\log_{2\sqrt{3}} 1728 = y$, then
$$\begin{align} (2\sqrt{3})^y &= 1728\\
2^y(\sqrt3)^y &= 1728\\2^y(3^\frac12)^y &= ...
2
votes
2answers
144 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}}) - ...
1
vote
1answer
42 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 ...
0
votes
2answers
54 views
solving in x involving both exponential and logarithmic function
Is it possible to solve a function with both exponential and logarithm such as
$a x^2−b.\log(x)= c$
in closed form; where $a,b,c$ are constants and $a>0$ and $b>0$?
0
votes
4answers
129 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
votes
1answer
43 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 ...
2
votes
4answers
123 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 ...
1
vote
2answers
36 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 ...
0
votes
2answers
70 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 ...
0
votes
1answer
14 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):
$81^{(\log_{3}N)+(log_{9}N)}$
...
2
votes
1answer
35 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 ...
2
votes
0answers
29 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 ...
1
vote
1answer
32 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?
1
vote
2answers
49 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 ...
1
vote
3answers
96 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.
1
vote
5answers
67 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 ...
0
votes
1answer
28 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.:
...
1
vote
1answer
27 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 ...
3
votes
3answers
52 views
Finding the derivative of a function with a Natural Log.
I am to differentiate the above function. I would think step one is to use the quotient rule of natural log expanding the expression.
However doing this would still leave $\ln(3x \tan(x)) - ...
1
vote
1answer
48 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 ...
8
votes
0answers
220 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
votes
1answer
55 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} ...
5
votes
1answer
181 views
Explain this code to compute $\log(1+x)$
It's well known that you need to take care when writing a function to compute $\log(1+x)$ when $x$ is small. Because of floating point roundoff, $1+x$ may have less precision than $x$, which can ...
3
votes
1answer
48 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 ...
2
votes
2answers
96 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
vote
1answer
32 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 ...
0
votes
1answer
27 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
5answers
296 views
Convergence of series $\sum_{n=1}^\infty \ln\left(\frac{2n+7}{2n+1}\right)$?
I have the series
$$\sum\limits_{n=1}^\infty \ln\left(\frac{2n+7}{2n+1}\right)$$
I'm trying to find if the sequence converges and if so, find its sum.
I have done the ratio and root test but It ...
1
vote
0answers
190 views
Question about $f_n=f_{n-1}+\ln f_{n-1}$ with $f_0=2$ [closed]
Let $n,m$ be strictly positive integers.
Let $f_0 = 2$. Let $f_n=f_{n-1}+\ln f_{n-1}$.
Let $h_{n,1}=\sinh^{-1}\left(\dfrac{n}{2}\right)$ and $h_{n,m}=\sinh^{-1}\left(\dfrac{h_{n,m-1}}{2}\right)$.
...
0
votes
4answers
110 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 $a,b>1$ ?
0
votes
1answer
28 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}) + ...
1
vote
1answer
44 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.
...
1
vote
3answers
109 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
63 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
115 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
votes
4answers
52 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
36 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
69 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
...
0
votes
2answers
68 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:
$$ ...
1
vote
2answers
48 views
Limit, log rule
I have couple of question on this part of equation -
\begin{align}
\lim_{n\to \infty } \frac{ 7 \cdot \sqrt{n}}{\log(n)}- \lim_{n\to \infty} \frac{1}{n\cdot \log(n)} &=\lim_{n\to \infty} \frac{7 ...
2
votes
2answers
69 views
Derivative of ${ x }^{ x }$ without logarithmic differentiation
With logarithmic differentiation, it is quite simple to compute the derivative of ${ x }^{ x }$:
$y={ x }^{ x }\\ \ln { y } =x\ln { x } \\ \frac { 1 }{ y } \frac { dy }{ dx } =\ln { x } +1\\ \frac { ...
1
vote
2answers
54 views
Finding the coefficient of $ x^n $ in the expansion of $ { ({\ log_e (1+x) })^2 } $
I've been trying to find the coefficient of $x^n$ in the expansion of $ { ({\log_e (1+x) })^2 } $.I wrote out the expansion of $ { ({\log_e (1+x) })^2 } $ explicitly and tried to generalize the terms ...
1
vote
3answers
55 views
How should I calculate the Lebesgue integral of logarithm function from zero to infinity?
Does the area under the $\ln(x)$ in $(0,+\infty)$ is measurable?
If yes, how can I calculate it?
0
votes
1answer
157 views
How to convert log10 values to decimal
I need to convert $\log_{10}(1.07366)$ to decimal.
Need the equation for the same.
6
votes
1answer
69 views
What is $ \lim_{x \to 0} \log_0(x) $?
As per the title; what is $ \lim_{x \to 0} \log_0(x) $ ?
According to WolframAlpha: $$ \lim_{x \to 0} \log_0(x) = 0 $$ but how is this possible?
Surely the limit should be indeterminate since ...
