6
votes
3answers
141 views

Something about $\frac{\log x}{x}$

Denote $\log x = \log_ex$. Let's consider the below function $$\frac{\log x}{x}$$. Apparently, It's maximum is $\frac{1}{e}$. and strictly increasing in $(0,e]$, strictly decreasing in $[e,+\infty)$. ...
0
votes
1answer
94 views

Unable to comprehend a connection between two equations

I was reading this paper and got stuck at the transition from Equation (13) to Equation (14) (p. 16/17). We got a function of the form: $y(t)=k(t)^{\alpha}h(t)^{\beta}$ We know it grows from zero ...
3
votes
1answer
71 views

On the equation $\exp(a x+b)=\ln(x)$

I am confronted with: $$\exp(a x+b)=\ln(x)$$ for $a,b$ reals and $a<0$, $b>0$. I need the (unique) solution for $x$. My first target is (if it exists) an analytic solution in terms of ...
0
votes
1answer
41 views

Proof of generalization of a particular limit converging to $e^{\frac{1}{(p-1)^2}}$

I was reading a very old and long article on logarithms in a library it has pages turned yellow and had one pages titled - Tricky problems I managed to solve 5 out of the 6 but I couldn't do this 6th ...
4
votes
3answers
102 views

Does $\operatorname{Log}(1+i)^2 =2\operatorname{Log}(1+i)$

And similarly, does $\operatorname{Log}(1-i)^2=2\operatorname{Log}(1-i)$? If we were dealing with real numbers, it would hold. But I'm guessing that the fact that there are imaginary numbers involved ...
5
votes
2answers
299 views

Definition of a logarithm

My question is as follows: Is the below a useful elementary way of dealing with negative arguments? If not, what is a better (elementary or not) way of dealing with negative arguments of the ...
2
votes
1answer
61 views

nth Root of a Rational Function

Suppose I have two polynomials $p(z)$ and $q(z)$ and a positive integer $n$. Suppose I wanted to define $r(z)=(\frac{p(z)}{q(z)})^{1/n}$ on $\Omega$ such that r(z) was analytic and single valued. On ...
0
votes
3answers
750 views

Upper Bound of Logarithm

For $1\leq x < \infty$, we know $\ln x$ can be bounded as following: $\ln x \leq \frac{x-1}{\sqrt{x}}$. Then what is the upper bound of $\ln x$ for following condition? $2\leq x <\infty$
2
votes
3answers
68 views

Evaluating $\lim\limits_{x\to 0^{+}} \frac{x}{\ln^2 x}$

How can I find: $$\lim_{x\to 0^+} \frac{x}{\ln^2 x} $$ I know that the limit is $0$. I tried sandwich theorem but I don't know what could be bigger. Thanks in advance.
6
votes
2answers
205 views

Formula for Sum of Logarithms $\ln(n)^m$

As you know $\sum_{n=1}^k \ln(n) =\ln(k!)$ is there a formula for $\sum_{n=1}^k \ln(n)^m$?
4
votes
4answers
146 views

How to find the limit of $\dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$?

How do you find $$\lim_{n \to\infty} \dfrac{\ln(\ln(\frac{n}{n-1}))}{\ln(n)}$$ I know it's $-1$, but I had to plot it.
1
vote
2answers
561 views

Natural logarithm limit

Is $$\lim_{n\rightarrow +\infty}\ln\left(\frac{n+1}{n}\right)=0?$$ Because it is $\ln(1+\frac{1}{n})$ and $\frac{1}{n}$ tends to $0$, since $n$ tends to infinity, so the limit becomes ...
0
votes
1answer
36 views

Bounding below the difference of sums

I would like to bound below the following expression: $\lambda(m,n)=\sum\limits_{i = 1}^{m+n}\lg{i} - \sum\limits_{i = 1}^{m}\lg{i} - \sum\limits_{i = 1}^{n}\lg{i}$ by some expression that ideally ...
-1
votes
1answer
99 views

equivalance between two equations with log and exponential

$d > \sigma$ ... (1) $\exp^{-(\frac{d^2}{2\sigma^2})} < 10^{-0.5}$ ... (2) Is (1) <==> (2) true ? EDIT: > replaced by < in the 2nd expression
1
vote
1answer
163 views

How to calculate $\int_{|z|=r}\ln(1-z)\,dz$ in dependence of $r\neq1$?

With the integration I mean one counter-clockwise turn around the origin, i.e. $$\int_{\phi=0}^{2\pi}\ln(1-re^{i\phi})ire^{i\phi}d\phi$$ For $r<1$, this is simply a contour integration on a ...
3
votes
3answers
360 views

Inequality for logarithms

I conjecture the following inequality is true $$\ln x \le (x - 1)\ln\frac{x}{x-1}$$ for all $x > 1$, but I cannot give a proof. I will appreciate if someone can provide one.
2
votes
1answer
152 views

Inequality involving $\log$

Let $g$ be a non-negative measurable function on $[0,1]$. How can I show that $$ \int \log ~(g(u))~\text{d}u \leq \log~\int g(u)~\text{d}u $$ whenever the left hand side is defined. If it helps, I ...
4
votes
1answer
417 views

Help understanding this formula on mutual information (used in bioinformatics)

I'm a bit lost on understanding this formula in my bioinformatics text, and I appreciate any tips or advice. Mutual Information, $\operatorname{MI}(X; Y)$ is: $$ \mu = \sum_x \sum_y p(xy) ...
0
votes
2answers
100 views

Why do I get different results using the e function for the ln?

I have a question which Kind of makes me crazy right now. If I have the equation $1=\ln(2•3)$, I could then use the e function to remove the $\ln.$ Doing that I get $e^1 = 6.$ Now say we are using the ...
2
votes
2answers
978 views

How to solve simple log inequality?

I've got $8n^2 \lt 64n\log(n)$ and I need to find the $n$ range if $n\gt 0$ to satisfy the inequality.
1
vote
1answer
64 views

Is possible to simplify $P = N^{ CN + 1}$ in terms of $N$?

Having: $P = N^{CN + 1}$; How can I simplify this equation to $N = \cdots$? I tried using logarithms but I'm stucked... Any ideas?