Questions related to real and complex logarithms.
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 ...
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
3answers
94 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
66 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.:
...
14
votes
5answers
497 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 ...
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
48 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
47 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
149 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
votes
1answer
54 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} ...
8
votes
0answers
218 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
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 ...
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 ...
2
votes
2answers
94 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
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
2answers
48 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 ...
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
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 $h_{n,m}=\sinh^{-1}\left(\dfrac{h_{n,m-1}}{2}\right)$.
...
1
vote
3answers
107 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
62 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:
$$ ...
1
vote
2answers
48 views
Limit, log rule
I have couple of question on this part of equation -
\begin{align}
\lim_{n\to \infty } \frac{ 7 \cdot \sqrt{n}}{\log(n)}- \lim_{n\to \infty} \frac{1}{n\cdot \log(n)} &=\lim_{n\to \infty} \frac{7 ...
2
votes
2answers
69 views
Derivative of ${ x }^{ x }$ without logarithmic differentiation
With logarithmic differentiation, it is quite simple to compute the derivative of ${ x }^{ x }$:
$y={ x }^{ x }\\ \ln { y } =x\ln { x } \\ \frac { 1 }{ y } \frac { dy }{ dx } =\ln { x } +1\\ \frac { ...
1
vote
2answers
54 views
Finding the coefficient of $ x^n $ in the expansion of $ { ({\ log_e (1+x) })^2 } $
I've been trying to find the coefficient of $x^n$ in the expansion of $ { ({\log_e (1+x) })^2 } $.I wrote out the expansion of $ { ({\log_e (1+x) })^2 } $ explicitly and tried to generalize the terms ...
0
votes
3answers
58 views
Question about changing a logarithm's base
I've been using the following method to derive/remember the logarithm base conversion formula: If I want to convert $\log_a(x)$ to an expression in base $b$, I say,
$$a^{\log_a(x)}=x\\
...
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?
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 ...
1
vote
1answer
41 views
Expansion of Lambert $W$ for negative values [duplicate]
What is a good approximation for the Lambert $W(x)$ function for values between $\frac{-1}{e}$ and $0$? Is it simply $x-x^2$? If so, what bounds are there on the error?
1
vote
3answers
56 views
Logarithm simplification
Simplify: $\log_4(\sqrt{2+\sqrt{3}}+\sqrt{2-\sqrt{3}})$
Can we use the formula to solve this: $\sqrt{a+\sqrt{b}}= \sqrt{\frac{{a+\sqrt{a^2-b}}}{2}}$
Therefore first term will become: ...
5
votes
2answers
56 views
Series involving log $\sum_{n=1}^{\infty} \left( n\log \left(\frac{2n+1}{2n-1}\right)-1\right) = \frac{1-\log 2}{2}$
Does anybody know how to prove this series?
$$\sum_{n=1}^{\infty} \left( n\log \left(\frac{2n+1}{2n-1}\right)-1\right) = \frac{1-\log 2}{2}$$
I arrived at this through Mathematica.
I tried writing ...
2
votes
0answers
104 views
Log-concave functions whose sums are still log-concave: possible to find a subset?
Rationale:
I am puzzled by a problem of log-concavity, which arises in population dynamics where the curvature of the logarithm of sums is a quantity of interest.
It is well-known that sums of ...
4
votes
2answers
59 views
Given $1<a<b<c$ prove $\log_a\log_ab+\log_b\log_bc+\log_c\log_ca>0.$
Given $1<a<b<c$ prove $$ \log_a\log_ab+\log_b\log_bc+\log_c\log_ca>0. $$
How to approach problems like this? I tried usual transformations but no help. I guess I have to use ...
0
votes
2answers
67 views
Simplifying a Logarithmic Expression
This is a really, very simple question, but I've never been an extremely confident mathematician and I just want to make sure that my attempt was correct. Oh and this is homework incase you were ...
4
votes
3answers
58 views
How to solve infinitely nested logarithms
I have an iterative process that starts with
$$x_1 = \log_{10}(a)$$
Following iterations are as follows:
$$x_2 = \log_{10}(a-b\cdot x_1)$$
$$x_3 = \log_{10}(a-b\cdot x_2)$$
$$x_4 = ...
2
votes
1answer
152 views
Evaluating $f(z)=\sqrt{z^2-1}$, given the branch I am on.
I'm working on a problem in Gamelin's Complex Analysis (Chapter IV, Section 2, page 109, exercise #4). I'm asked to consider the branch of $f(z)=\sqrt{z^2-1}$ on $D=C\setminus (-\infty,1]$ that is ...
0
votes
0answers
32 views
How to formulize this logarithmic expression?
If you have this list of numbers {$1, 2, 4, 7, 11, 16, 21, 27, 33, ...$}
Where the list starts at 1, then the next number is $1 + x + \left\lceil\log_2{x}\right\rceil$ where $x$ is the current ...
1
vote
0answers
40 views
Logarithmic simplification of a sum of power terms
It is entirely possible that there is no solution to this problem, but here goes...
I have a number of equation of states for fluids that have terms that are of the form
${\phi _r} = ...
2
votes
2answers
105 views
Solving equation $A^{(B^{(C^x)})} = C^{(B^{(A^x)})}$ for $x$
Recently I came across the equation
$$A^{(B^{(C^x)})} = C^{(B^{(A^x)})}$$
where $A \neq B \neq C$, and if $A, B, C > 1$ or if $0 < A,B,C < 1$, there exists
a unique solution for $x$.
Here ...
0
votes
2answers
214 views
How to enter subscript characters in WolframAlpha? [closed]
I'm trying to enter equations like this in WolframAlpha.
How do I format this?
3
votes
2answers
49 views
Combination of logarithms and exponents
I am given this question and told to solve for $a,b,c$:
$$\frac{y^{8a}x^{b}\log_x(y^{8a})}{2x^2y^c} = \frac{y^{3/2}\ln(y)}{3\ln(x)}$$
I tried to convert all the logarithms to $\ln$ and remove the ...
0
votes
1answer
23 views
If $f(x) = \log_ax$, show that $\frac{f(x+h)-f(x)}{h} = \log_a(1+\frac{h}{x})^{1/h}$
If $f(x) = \log_ax$, show that
$$\frac{f(x+h)-f(x)}{h} = \log_a\left(1+\frac{h}{x}\right)^{1/h},$$
where $h\neq0$.
1
vote
2answers
114 views
Intuition behind logarithm inequality
One of fundamental inequalities on logarithm is:
$$ 1 - \frac1x \leq \log x \leq x-1 \quad\text{for all $x > 0$},$$
which you may prefer write in the form of
$$ \frac{x}{1+x} \leq \log{(1+x)} \leq ...
0
votes
2answers
91 views
Find $n$ in $8n^2 \le 64n\lg n$
Given the solution.
Can someone help me why $n \le 43$. What is the step by step of the solution for this?
