Tagged Questions
0
votes
1answer
37 views
Looking for suggestions on how to proceed with showing that:
for $x \ge 2863:$
$$\ln\left(\left\lfloor\frac{x}{6}\right\rfloor!\right) < \sum_{k=5}^{\infty}-\mu(k)\ln\left(\left\lfloor\frac{x}{k}\right\rfloor!\right)$$
I've written a java application which ...
15
votes
1answer
201 views
$x^3-3x-3=0$, prove that $10^x<127$
$x$ is the real root of the equation $$3x^3-5x+8=0,\tag 1$$ prove that $$e^x>\frac{40}{237}.$$
I find this inequality in a very accidental way,I think it's very difficult,because the actual value ...
0
votes
0answers
58 views
Using Stirling's Formula to approximate a difference of logarithms of factorials in the same way as Jitsuro Nagura.
In Jitsuro Nagura's classic proof of a prime existing between $x$ and $\frac{6x}{5}$, he uses Stirling's formula to show that:
$$T\left(x\right) - T\left(\frac{x}{2}\right) - ...
0
votes
0answers
41 views
Do these inequalities regarding the gamma function and factorials work?
I am seeing if it is possible to generalize lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$.
In a previous question, I asked whether the following inequality is ...
2
votes
0answers
55 views
Trying to generalize an inequality from Jitsuro Nagura: Does this work?
I am investigating the generality of Lemma 1 in Nagura's proof that there is always a prime between $x$ and $\frac{6x}{5}$.
In Lemma 1, Nagura establishes when $n > 1$, $x \ge 1$:
...
0
votes
1answer
25 views
How do you evaluate an inequality that involves logarithms of factorials?
For $x > 1$, $n > 2$ with $2 \mid x+1$ and $n \mid x+1$, does it then follow that:
$$\log(\lfloor\frac{x}{2}\rfloor!) - \log(\lfloor\frac{x}{n}\rfloor!) \le \log(\lfloor\frac{x+1}{2}\rfloor!) - ...
33
votes
3answers
609 views
Prove $(\dfrac{2}{5})^{\frac{2}{5}}<\ln{2}$
Inadvertently, I find this interesting inequality,But this problem have nice solution?
prove that
$$\ln{2}>(\dfrac{2}{5})^{\frac{2}{5}}$$
This problem have nice solution? Thank you.
ago,I find ...
0
votes
0answers
31 views
Does it follow that if $\{\frac{x}{2}\} \ge \frac{1}{2} + \frac{\{x\}}{2}$, $\log(\lfloor\frac{x}{2}\rfloor!) \le \log\Gamma(\frac{x+1}{2})$?
The answer seems to be yes.
Here's my reasoning.
Let $\{x\} = x - \lfloor{x}\rfloor$
Assume $\{\frac{x}{2}\} \ge \frac{1}{2} + \frac{\{x\}}{2}$
$$\log(\lfloor\frac{x}{2}\rfloor!) = ...
5
votes
1answer
98 views
Understanding a famous proof by Jitsuro Nagura: Need help understanding one step in the main theorem
I am going through the proof by Jitsuro Nagura which shows that there is always a prime between $x$ and $\frac{6x}{5}$ where $x \ge 25$.
Nagura uses the following definitions:
$$\vartheta(x) = ...
8
votes
1answer
205 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 ...
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
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 $h_{n,m}=\sinh^{-1}\left(\dfrac{h_{n,m-1}}{2}\right)$.
...
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 ...
1
vote
2answers
110 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
$$ \frac{x}{1+x} \leq \log{(1+x)} \leq ...
1
vote
2answers
94 views
Proof of inequality involving logarithms
How could we show that
$$\left|\log\left( \left({1 + \frac{1}{n}}\right)^{n + \frac{1}{2}}\cdot \frac{1}{e}\right)\right| \leq \left|\log\left( \left({1 - \frac{1}{n}}\right)^{n - \frac{1}{2}}\cdot ...
1
vote
1answer
66 views
How to prove: for some $c>0,x>2 , c,x\in \mathbb R , \, \int_2^x \frac{\mathrm dt}{\log t}-\frac{x}{\log x} \leq \frac{cx}{(\log x)^2}$
How to prove:
for some $c>0,x>2 , c,x\in \mathbb R$
$$ \int_2^x \frac{\mathrm dt}{\log t}-\frac{x}{\log x} \leq \frac{cx}{(\log x)^2}$$
I have tried my textbook, notes and also tried to find ...
1
vote
1answer
44 views
A simple inequality with logarithms and exponential
I want to prove that for $k>0$:
$ 2^k \geq \frac{-1}{\log_2(1-\frac{1}{2^k})}$
I've plotted both functions and it seems to be the case for k>0.
In fact, it would also be nice to see that:
$ ...
2
votes
2answers
103 views
About the use of Stirling approximation
How to prove this inequality:
$$\ln \Gamma \left( x \right)-2\ln \Gamma \left( \frac{x+1}{2} \right)>\frac{2x}{3}$$
Sry I forgot to mention that $x>300$
3
votes
3answers
52 views
Solving inequality involving logarithms
I must be doing something wrong. I want to solve the following, where n is a positive integer, and p is a real number between 0 and 1.
$$(1-p)^n \le 0.4$$
So I take the log on both sides:
...
5
votes
3answers
102 views
Why is $\int\limits_{1}^{n} \log x \,dx \le \sum\limits_{x = 1}^{n}\log x$?
It has been a long time since I studied integrals, so this question may sound stupid. I was going through this wiki page, and came across the following inequality:
$$\int_{1}^{n} \log x \,dx \le ...
4
votes
2answers
127 views
I am trying to solve the inequality $\log_{\log{\sqrt{9-x^2}}} x^2 <0$
I am trying to solve the inequality
$$\log_{\log{\sqrt{9-x^2}}} x^2 <0.$$
I got $\mathrm{S.S}=(-\sqrt8 ,-1)\cup( 1,\sqrt8)$, but a friend got $\mathrm{S.S}=(-1,1)- \{0\}$.
Please, what is ...
1
vote
2answers
54 views
Explanation of this inequality
Is there a graphic visualization of $\sum_{k=1}^{n} 1/k \, \, \leq \, \, \,1 \, + \, \int_1^n \! (1/x) \, \mathrm{d} x$ as intuitive as the integral test ?
I can't see why the inequality is true.
I ...
2
votes
1answer
64 views
Simple inequality help
I need a function $f(x)$ that satisfies the properties bellow for all integers $k$
$$ \frac{\log(k+1)}{k+1}-\log\left(1+\frac 1 k\right)+f(k+1)-f(k)<0 \ $$
$$ \lim_{k \rightarrow \infty} f(k)=0 $$
...
4
votes
1answer
78 views
Trouble proving ln inequality
I looking for help with proving the following inequality. Any relevant logarithmic identities would be great. Tried differentiating and taking limits and I'm lost as to how to approach this.
$$\frac ...
16
votes
6answers
427 views
$\log_9 71$ or $\log_8 61$
I am trying to know which one is bigger :$$\log_9 71$$ or $$\log_8 61$$ how can i know without using a calculator ?
3
votes
1answer
119 views
How to prove $\left\|\ln\left(e^{iH_1}e^{iH_2}\right)\right\|\leq\left\|H_1\right\|+\left\|H_2\right\|$?
Let $H_1$ and $H_2$ denote arbitrary Hermitian operators (finite dimensional) and let $\left\|\ldots\right\|$ denote the usual operator norm. I conjecture that
$$
...
1
vote
1answer
126 views
Logarithm in an inequality: is it solvable?
Can anyone help me understand what happens to the following inequality once I apply a logarithm to all three parts?
$$
- \varepsilon < 2^{\frac{1}{x}} < \varepsilon \longrightarrow \ln{(- ...
3
votes
4answers
250 views
Prove $\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$
Prove $$\log_{\frac{1}{4}} \frac{8}{7}> \log_{\frac{1}{5}} \frac{5}{4}$$
How to prove without a computer?
3
votes
3answers
145 views
How to prove this ln inequality?
I have the following inequality, which (supposedly) holds for every $x\in\mathbb{R}$:
$$
1+x\ln\left(x+\sqrt{1+x^{2}}\right)\geq\sqrt{1+x^{2}}
$$
I've been struggling to find some known inequalities ...
1
vote
2answers
94 views
Prove the statement : $\log(k + 1) -\log k>\frac{ 3}{10k}$
Prove the statement : $\log(k + 1) - \log k > \frac{3}{10k}$
Approach :
$$\log(k+1)-\log{k} > \frac{3}{10k}$$
Clearly, $k\in\mathbb{Z}^{+}$
...
2
votes
4answers
87 views
Show these simple inequalities
Show that $(\log(1+x))^2\le x$ and that $(\log(1+x))^2\le x^2$ for all $x\ge0$.
Both of these inequalities seem to be true, judging from plotting these functions with a grapher.
Can you help me ...
9
votes
4answers
357 views
How to know if $\log_78 > \log_89$ without using a calculator?
I realize that I lack any numerical intuition for logarithms. For example, when comparing two logarithms like $\log_78$ and $\log_89$, I have to use the change-of-base formula and calculate the values ...
3
votes
3answers
213 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.
3
votes
3answers
132 views
Squeeze an integral
Would you have any idea about this problem ?
Prove that for all nonnegative integers $n$, the following inequalities hold:
$$\frac{e^2}{n+3}\leq \int_{1}^{e} x (\ln x)^n \,dx \leq ...
1
vote
1answer
126 views
Inequality involving $\log$
Let $g$ be a non-negative measurable function on $[0,1]$. How can I show that $$ \int \log ~(g(u))~\text{d}u \leq \log~\int g(u)~\text{d}u $$ whenever the left hand side is defined.
If it helps, I ...
1
vote
1answer
89 views
Why is the following about logarithms true?
I was reading some algorithm's analysis and I came across the following in the proof:
$\log_2(n+1) \le h \le 1 + \log_2(n) \implies h = \lceil \log_2(n+1)\rceil$
Here both $h$ and $n$ are integral. ...
4
votes
1answer
237 views
Comparing Powers of Different Bases
How can I know if one power is bigger than the other when the bases are different?
For example, considering $2^{10}$ and $10^{3}$ the former is the greater one, but how to prove this? Logarithms? ...
6
votes
1answer
96 views
logarithms power equation
I got a home work question to solve the following:
$$ 27x^2 < x^{\log_3x} $$
can any one please explain how to solve this type of equation?
I have no idea what to do or what to search for.
1
vote
1answer
209 views
How do you prove the following inequality concerning complex Logarithms?
If $0<|w|<1/2$, then $2|w|/3<|\operatorname{Log}(1+w)|$ using power series and modulus inequalities.
0
votes
2answers
67 views
Trouble understanding how an equality is obtained
(This is from a proof by contradiction, so that's why the equality does not actually hold. Edited for brevity; I don't think I've omitted anything pertinent to my questions.)
[...] The ...
4
votes
1answer
119 views
Show the correctness of a logarithmic inequality
Let $p_1>p_2$ and $n_1>n_2$ be positive numbers. I want to show that,
$$
\frac{\log \left(\frac{p_1}{n_1}+1\right)}{\log \left(\frac{p_2}{n_2}+1\right)}\leq \frac{\log ...
1
vote
4answers
103 views
Solving equality to find upper limit
I need to find a sensible upper limit for a part of an algorithm in a program I am writing. I have boiled it down to this.
Given $a$, $b$ and $c$, find $x$ in $a^{x-1}b < c < a^{x}b$.
But I ...
4
votes
3answers
528 views
How to solve $n < 2^{n/8}$ for $n$?
This is from an exercise (1.2.2) in introduction to algorithms that I'm working on privately.
To find at what point a $n \lg n$ function will run faster than a $n^2$ function I need to figure out for ...
2
votes
2answers
612 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.
