1
vote
2answers
74 views

$-\varepsilon\log(x)\overset{?}{\geq} -\log(\varepsilon x)$

I'm refering to this proof: http://en.wikipedia.org/wiki/Quantum_relative_entropy#The_result In there it's stated that "Since the matrix $(P_{ij})_{ij}$ is a doubly stochastic matrix and $-\log$ is a ...
8
votes
2answers
232 views

A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$ \int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*) $$ while evaluating this log-cosine integral. I made several ...
8
votes
2answers
309 views

A series with only rational terms for $\ln \ln 2$

We all know that $$ \ln 2 = \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n}. $$ Do you know a series with only rational terms for $$\ln \ln 2 = ?$$ Let's exclude base expansions with non ...
2
votes
0answers
25 views

Equivalence of criteria using logarithmic transformation

Is the following criterion: $$ \frac{\partial^2 f}{\partial x\partial y} = \frac{\partial^2 f}{\partial y\partial x} $$ Equivalent to: $$ \frac{\partial^2 \ln f}{\partial x\partial y} = ...
2
votes
5answers
126 views

How I could show that :$\log1=0$?

I would be like somone to show me or give me a prove for this : Why $\ln 1=0$ ? Note that $\ln$ is logarithme népérien, the natural logarithm of a number is its logarithm to the base $e$. Thanks ...
0
votes
0answers
26 views

Prove natural log between two finite harmonic sums [duplicate]

Prove for n in the naturals we have: $$\sum_{k=2}^n 1/k \le \ln(n) \le \sum_{k=1}^{n-1} 1/k$$ Intuitively this makes sense to me but I can't for the life of me figure out how to start this proof.
1
vote
3answers
68 views

Convergence N'th Harmonic number minus the Natural Logarithm of N. [duplicate]

I was hoping if someone could show me the proof of exactly why this converges to the Euler–Mascheroni constant.
5
votes
3answers
60 views

Limit of logarithmic function using l'Hospital

How can I find the following limit: $$\lim_{x\rightarrow \infty}\frac{\ln(1+\alpha x)}{\ln(\ln(1+\text{e}^{\beta x}))}$$ where $\alpha, \ \beta \in \mathbb{R}^+$. My first guess was to use ...
3
votes
1answer
77 views

Tighter logarithmic inequality

There is a well-known lower bound for $$ x\log{1+x\over x}\geq {x\over1+x} $$ for $x\geq0$. I know a tighter lower bound on the same domain $$ x\log{1+x\over x}\geq{2x\over1+2x}\geq {x\over1+x}. $$ It ...
3
votes
1answer
46 views

Could somebody validate my proof regarding the limit of $\ln(x_n)$ when, $x_n$ tends to $a$?

So, let me cearly state the problem: Let $(x_n)$ be a convergent sequence, with: $ x_n > 0 $, $\forall n$, n natural number, and $x_n \to a$, with $a>0$. Then $\ln{x_n} \to \ln{a}$. Here is my ...
1
vote
4answers
87 views

Prove that $\ln x \leq x - 1$

I need help with this proof for my real analysis class. it is on the practice sheets and we do NOT get an answer. I proved $\ln(x) < x−1$ for all $x>1$ by contradiction but cannot do this one. ...
2
votes
0answers
26 views

Yacov Perelman Nepero game

This is my first question, so sorry if I'll make any mistake in using the site formatting. I found this game on a book by Yacov Perelman and I thought it could be nice to introduce Nepero number to ...
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
2answers
61 views

Logarithmic limit proof

This has been described as another "way" to do "logarithmic" limits. a. Given that $a^n=e^{n \ln a}$ prove that if $0<a<1$, then $\lim_{n\to \infty}a^n = 0$ This intuitively makes sense but I ...
0
votes
2answers
51 views

How to solve equation

I have a problem to solve that equation $\log_{4}\left(x\right) = -2x + 9$ i know that answer is 4 but what is step by step solution.
1
vote
1answer
58 views

Growth of |logx| versus of 1/x

Do you think there is a number k s.t. $\int_{(0,\infty)} \frac{|log(x)|^{k}}{x}d\mu$ will converge,where $\mu$ is the Lebesgue measure? If you don't know ,can you at least give me some reference for ...
2
votes
4answers
122 views

Showing $\frac{x}{1+x}<\log(1+x)<x$ for all $x>0$ using the mean value theorem

I want to show that $$\frac{x}{1+x}<\log(1+x)<x$$ for all $x>0$ using the mean value theorem. I tried to prove the two inequalities separately. $$\frac{x}{1+x}<\log(1+x) \Leftrightarrow ...
1
vote
6answers
76 views

Noncircular construction of $e$ and $\ln$ for the real line

Could anyone direct me to (or possibly detail) a construction of $e$ and $\ln$ along the reals? For example, they can define $e=\lim_{n\rightarrow\infty}(1+\frac{1}{n})^n$ but from this definition ...
1
vote
0answers
14 views

Trying to prove a function is increasing in one variable after the second variable crosses a particular value.

Let $f(p,n)$ = $\ln\bigl(1 + \frac{ap}{b+I}\bigr) + \sum_{j = 1}^{n}\ln\bigl(1+ \frac{b_j}{b + I - b_j + ap}\bigr)$, where $b>0$ and $a>0$ are fixed constants. The positive real numbers $b_j$ ...
2
votes
3answers
100 views

Proof $\lim\limits_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$

I want to show that for all $a \in \mathbb{R }$ $$\lim_{n \rightarrow \infty}n(a^{\frac{1}{n}}-1)=\log a$$ So far i've got $\lim\limits_{n \rightarrow \infty}ne^{(\frac{1}{n}\log a)}-n$, but when i ...
2
votes
1answer
67 views

Baby Rudin Excercie 1.7 (Existence of Logarithms)

I'm currently working through Baby Rudin and need help on exercise 1.7(f). The question: Let $A$ be the set of all $w$ such that $b^w < y$, and show that $x = \text{sup}(A)$ satisfies $b^x = ...
2
votes
2answers
67 views

Why is $x\log(x)$ convex?

Why is $x\log(x)$ convex? According to the definition it must hold: $(tx+(1-t)y)\log(tx+(1-t)y)\le tx\log(x)+(1-t)y\log(y)$ for all positive $x,y$ and $t\in[0,1]$ edit: It is allowed to ...
1
vote
1answer
23 views

Show combination of affine functions and logs has at most one zero

For $x>0$, let $$ f(x)=(x+2){\sf log}(x)-(x+1){\sf log}(x+1) $$ Can anybody show that the equation $f(x)=0(x > 0)$ has at most one solution.
0
votes
0answers
30 views

Hessian of a conic function

i got a conic System: $Ax =b, x\in C$, where $A\in\mathbb{R}^{m\times n}, b\in\mathbb{R}^m$ and C is the cone of the $n\times n$ positive semidefinite matrices, so ...
0
votes
2answers
59 views

Limit of a harmonic subseries minus “its” logarithm

$\displaystyle \lim_{n \to \infty} \sum_{k=1}^n \frac{1}{3k-1} - \frac{1}{3}\ln(n)$ I think that inserting the other terms and then subtracting them would not help. I need just the ideea. Thank you.
0
votes
2answers
28 views

Inequality for negative logarithms?

Given $0 < x < y < 1$, is it possible to prove the following result: $\frac {ln\:x}{ln\:y} < 1$? Thanks
1
vote
1answer
45 views

At which parameter value $c>0$ do the number of solutions of $\log(1+x^2)=x^c$ change?

I'm looking at the functions $x\mapsto \log(1+x^2)$ and $x\mapsto x^c,\ c>0$ on the interval $\mathbb R^+_0$. I'm interested in the properties of $$\log(1+x^2)=x^c.$$ Graphically, for small $c$, ...
3
votes
2answers
122 views

Alternatives to percentage for measuring relative difference?

If the value of something changes from $\;a\;$ to $\;b\;$, their relative difference can be expressed as a percentage: $$ \newcommand{\upto}{\mathop\nearrow} (D0) \;\;\; a \upto b \;=\; (b-a)/a ...
25
votes
1answer
462 views

Power towers: to infinity and all the way back

In the following, let $n$ be a positive integer, all other variables be real (furthermore, $a>1$), all functions be real-valued, and logarithms of negative arguments be undefined. Let ...
6
votes
3answers
105 views

Need help with logarithmic differentitation

I have the expression $$y = \sqrt{x^2(x+1)(x+2)}.$$ I have tried looking at videos but I still cannot arrive at the correct answer and don't know how to get there. By the way, the correct answer is ...
2
votes
1answer
48 views

Generalised logaritmic function

I was wondering if there was a function that extends the domain of the following function to non-negative real numbers. For non-negative integer $n$ and real $y$, $y = f(x,n)$ is given by: $$f(x,n) = ...
3
votes
3answers
124 views

If $z_n \to z$ then $(1+z_n/n)^n \to e^z$

We are dealing with $z \in \mathbb{C}$. I know that $$ \left(1+ \frac{z}{n} \right)^n \to e^{z} $$ as $n \to \infty$. So intuitively if $z_n \to z$ then we should have $$ \left(1+ \frac{z_n}{n} ...
1
vote
2answers
56 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 ...
6
votes
1answer
77 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 ...
0
votes
2answers
107 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?
1
vote
2answers
257 views

Find the limit of the sequence containing logarithm??

Find $\lim_{n→∞} [log(2+3^n)]/2n$ I have my work till the very last step then i dont know how to continue $\lim_{n→∞} [log(2+3^n)]/2n$ =$\lim_{n→∞} log(3^n)+\lim_{n→∞} log[(2+3^n)/3n]$ ...
0
votes
1answer
86 views

Is $\lim_{x \to x_0} \log(f(x)) = \log\lim_{x \to x_0} f(x)$ always true?

This property is always true? If yes I would like a proof, otherwise an counterexample. $$\lim\limits_{x \to x_0} \log(f(x)) = \log\lim\limits_{x \to x_0} f(x)$$
3
votes
2answers
116 views

Evaluate: $\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}$

Evaluate: $$\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}$$ attemp: Take $P=\lim_{n \to \infty}[(1+\frac{1}{n})^n-(1+\frac{1}{n})]^{-n}$ . Then taking log both side .$$\ln ...
5
votes
2answers
406 views

Prove that $\log(1 + \sqrt{1+x^2})$ is uniformly continuous?

Problem Prove that $$\log(1 + \sqrt{1+x^2})$$ is uniformly continuous. My idea is to consider $|x - y| < \delta$, then show that $$|\log(1 + \sqrt{1+x^2}) - \log(1 + \sqrt{1+y^2})| = ...
1
vote
2answers
211 views

Interchanging limits and logarithms

This is probably not too smart, just wondering of the name of this rule: $$ \log \lim_{x \to x_0}f(x) = \lim_{x \to x_0}\log f(x) $$ A reference to a source and/or proof would be good too.
4
votes
1answer
143 views

What about $\exp((\log x)(\log y))$?

Is anything interesting known about the binary operation $$ x\circ y = \exp_b((\log_b x)(\log_b y)) $$ where $0<b\ne 1$? It's clearly commutatitive and associative, and satisfies $\forall x\in ...
5
votes
7answers
1k views

Is there any significance to the logarithm of a sum?

Many years ago, while working as a computer programmer, I tracked down a subtle bug in the software that we were using. Management had dispaired of finding the bug, but I pursued it in odd moments ...
6
votes
2answers
1k views

Is the natural log of n rational?

It's famously unknown whether the natural log of 2 is rational or not. How about the natural log of other numbers. Is it known/unknown whether these are rational? Obviously ln(1) is 0, and ln(2^n) ...