All Questions
Tagged with logarithms inequality
773
questions
3
votes
1
answer
195
views
Generalizing a logarithmic inequality
Let $\{x_i\}_{i=1}^N$ and $\{y_i\}_{i=1}^N$ be real numbers in the interval (0,1). Define for each $i$
$$\alpha_i = x_i (1-x_i) \log^2 \frac{x_i (1-y_i)}{y_i (1-x_i)}$$
and
$$\beta_i = x_i \log \frac{...
- 400
2
votes
1
answer
54
views
Prove $\ln (1+x) \leq x - x^2/4 $ for $x \leq 1$ using Taylor's theorem
RTP: $\ln (1+x) \leq x - \frac{1}{4} x^2$ for $x \leq 1$
I am trying to prove this specifically using Taylor theorem. Here is what I have so far:
$\ln (1+x) = x - \frac{x^2}{2} + \frac{x^3}{3(1+\xi)^3}...
- 487
1
vote
1
answer
78
views
How to show $(\log_2 x)^4 \leq x^3$ for $x > 1$?
In Rosen's discrete Math textbook, they mention in the solutions for one problem that $(\log_2 x)^4 \leq x^3$ for $x > 1$. However, I'm not sure how to exactly derive that myself, nor does the ...
- 883
0
votes
1
answer
46
views
Determining the witnesses (constants $C_0$ and $k_0$) when showing $(log_b n)^c$ is $O(n^d)$ (b > 1 and c,d are positive)
I'm having a hard time finding the constants/witnesses $C_0$ and $k_0$ that show $(\log_b n)^c$ is $O(n^d)$. That is $|(\log_b n)^c| \leq C_0|n^d|$ for $n > k_0$ (b > 1, and c,d are positive).
I ...
- 883
0
votes
6
answers
195
views
How would you prove $\log_{2}x < \sqrt x$ for $x > 16$? [closed]
I'm not really showing how to prove this, since I tried finding the $x$-intercepts/zeros of $f(x) = \sqrt x - \log_{2} x$ , and see that $x = 4, 16$ work but inspection, but I'm not sure how to ensure ...
- 883
0
votes
0
answers
21
views
Inequality for log-convex functions
Let $s:\mathbb{R}^+\to\mathbb{R}^+$ be a positive, decreasing, log-convex function such that $s(0)=1$, $\lim_{x\to\infty}s(x) = 0$. In addition, $-s'$ is also assumed to be log-convex.
Are these ...
- 93
3
votes
2
answers
86
views
logarithmic inequality $\log_x64 < 2$
I tried to solve this inequation and I got the solution.
So $\log_x2^6=6\log_x2$, then i divided by 6 both sides and I got $\log_x2<\frac{1}{3}$.
I multiplied by 3 and put it into logarithm and got ...
- 41
3
votes
2
answers
153
views
Logarithmic inequality involving $a_1, a_2, ..., a_n$
Given the real numbers $a_1, a_2,...,a_n$ all greater than $1$, such that $\prod_{i=1}^{n} a_i=10^n$, prove that:
$$\frac{\log_{10}a_1}{(1+\log_{10}a_1)^2}+\frac{\log_{10}a_2}{(1+\log_{10}a_1 + \log_{...
- 409
0
votes
4
answers
64
views
Have you seen this logarithmic inequality: $4\log(x)\log(y) \leq \log(xy)^2$
I have stumbled upon this following logarithmic inequality relating the product of two logs. For every $x,y > 0$
$$
4\log(x)\log(y) \leq \log(xy)^2.
$$
Furthermore, it holds as an equality if and ...
10
votes
2
answers
797
views
Question regarding nature of logarithmic equations
While reading my textbook's chapter about logarithms and seeing the solved examples I noticed in various places that the author was able to make the $\log$ just disappear in a equation or inequality ...
- 356
2
votes
4
answers
136
views
Solve the equation $\left(\frac{1+\sqrt{1-x^2}}{2}\right)^{\sqrt{1-x}} = (\sqrt{1-x})^{\sqrt{1-x}+\sqrt{1+x}}$
Solve in $\mathbb{R}$:
$
\left(\frac{1+\sqrt{1-x^2}}{2}\right)^{\sqrt{1-x}} = (\sqrt{1-x})^{\sqrt{1-x}+\sqrt{1+x}}
$
My approach:
Let $a = \sqrt{1-x}$ and $b = \sqrt{1+x}$ so $a^2 + b^2 = 2$. The ...
- 637
3
votes
1
answer
94
views
Prove that $ a^{\log_b(a)} + b^{\log_c(b)} + c^{\log_a(c)} \geq a^{\sqrt{\log_b(a)}} + b^{\sqrt{\log_c(b)}} + c^{\sqrt{\log_a(c)}} \geq a + b + c$
Let $ a, b, c > 1 $. Prove that
$ a^{\log_b(a)} + b^{\log_c(b)} + c^{\log_a(c)} \geq a^{\sqrt{\log_b(a)}} + b^{\sqrt{\log_c(b)}} + c^{\sqrt{\log_a(c)}} \geq a + b + c. $
My approach: I denoted $ \...
- 637
1
vote
1
answer
44
views
Bounding the solution of a logarithmic equation
Given a small number $\varepsilon >0$ and a constant $1/3\le \alpha < 1 $, I am looking for the smallest possible number $x^*$ such that for all real $x\ge \max\{x^*,3\}$, we have
$$\frac{x}{(\...
0
votes
1
answer
44
views
Finding the $x$ in the following logarithmic inequality
I am currently practising for my maths exam and I have come across this problem, which I can partially do:
$$\log^2_{1/2} x<4+\log_{1/2} x $$
After I put everything on the left hand side I get ...
- 35
2
votes
1
answer
53
views
Question about three lines theorem
I'm trying to prove that the function defined as $$F_\epsilon(z)=F(z)M_0^{z-1}M_1^{-z}e^{\epsilon z(z-1)},$$ where $F$ is an holomorphic function on $0<Re z<1$ and continuous and bounded on the ...
