All Questions
97
questions
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
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:
$$\frac{\log_{10}a_1}{(1+\log_{10}a_1)^2}+\frac{\log_{10}a_2}{(1+\log_{10}a_1 + \log_{...
- 409
10
votes
2
answers
797
views
Question regarding nature of logarithmic equations
While reading my textbook's chapter about logarithms and seeing the solved examples I noticed in various places that the author was able to make the $\log$ just disappear in a equation or inequality ...
- 356
0
votes
1
answer
67
views
Solutions to Some Logarithmic Inequalities
Suppose we have an inequation as shown below:$$I_0:\space \ln (x) > \frac{x-2}{x}$$ Now we would like to find the largest set $S$ of real numbers such that any element $p\in S$ will satisfy $I_0$ ...
4
votes
5
answers
595
views
Prove that $\ln x\leq\frac{x^{x+1/x}-1}{2}$ [ not solved ]
Prove that $$\ln x\leq\frac{x^{x+1/x}-1}{2}$$ is true for every positive real number, without calculus/derivative . (i.e. using some inequalities)
My progress. For $x\geq 1$ using $x+1/x\geq 2$ we ...
- 65
1
vote
2
answers
73
views
Prove that $\ln x\leq\frac{x^{\frac{x+\sqrt{x}+1}{2\sqrt{x}}}-1}{\sqrt{x}}$ for all positive real numbers
I am trying to prove the inequality $$\ln x\leq\frac{x^{\frac{x+\sqrt{x}+1}{2\sqrt{x}}}-1}{\sqrt{x}}$$ is true for all positive real numbers, without using calculus.
I realized the equality occurs if ...
- 65
-2
votes
1
answer
338
views
How do you solve: $\log_{\cos(x)} \sin(x) + \log_{\sin(x)} \cos(x) \le 2$ [closed]
I have the following inequation:
$$\log_{\cos(x)} \sin(x) + \log_{\sin(x)} \cos(x) \le 2$$
I know that $\sin(x)$ and $\cos(x)$ will give values in the interval $[-1, 1 ]$ but in the base there can't ...
- 1
7
votes
4
answers
447
views
How to prove $\frac{\sqrt{5}+1}{2}>\log_23$?
I want to compare the magnitude of $2\cos36^\circ$ and $\log_23$ without using calculators.
Using $\sin72^\circ=\cos18^\circ$, I can get $$4\sin18^\circ\left(1-2\sin^218^\circ\right)=1$$
then I can ...
- 331
-2
votes
1
answer
54
views
Given the inequalities $n > 2^k - 1$ and $n < 2^{(k+1)}$, express $k$ in terms of $n$ (where $k, n \in\Bbb N$). [closed]
I am quite certain that it will require logarithms and the floor function.
I haven't really attempted this on my own, because I have no idea how to even approach this problem. I have, however, spent a ...
- 11
10
votes
7
answers
897
views
How to prove that $\log_5(6)>\log_6(7)$?
I know that $\log_5(6)>\log_6(7)$ but I wanted to prove it without calculating the values.
After generalizing it it turned this way (for $x>1$):
$$\frac{\ln(x)}{\ln(x-1)}>\frac{\ln(x+1)}{\ln(...
- 111
6
votes
2
answers
239
views
Prove that: $\sqrt [3]{36}<\ln 28<\sqrt [3]{37}$
Prove that:
$$\sqrt [3]{36}<\ln 28<\sqrt [3]{37}$$
This inequality is the result of an integral representation/inequality.
I lost access to the article that mentioned this inequality. Now I ...
- 1,659
2
votes
2
answers
106
views
Domain of $h(x) = \log_2 (x^2+4)$
I am setting the $x^2+4 > 0$ and solving, but this leaves me with a question I can't seem to understand.
The resource I'm using is saying the domain is $(-\infty, \infty)$; $x$ is a set of all Real ...
- 135
0
votes
0
answers
43
views
Can this inequality be solved for $q$ in terms of $n$ (or the other way around), if $q^k n^2$ is an odd perfect number with special prime $q$?
Let $q^k n^2$ be an odd perfect number with special prime $q$ satisfying $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$.
The following general inequality relating $k$ and $q$ is proved in this ...
- 8,010
2
votes
1
answer
65
views
Find $m$ such that the inequality has $4$ integer solutions.
Find all values of $m$ such that the inequality: $\log(60x^2+120x+10m-10)>1+3\log(x+1)$ has exactly $4$ integer solutions?
The first thing I did was to gather from the inequality;
$\log(60x^2+120x+...
- 449
2
votes
2
answers
79
views
Prove $\log_{a}{(\frac{b^2}{ac}-b+ac})\cdot\log_{b}{(\frac{c^2}{ab}-c+ab})\cdot\log_{c}{(\frac{a^2}{bc}-a+bc})\geq1 $
Prove $$\log_{a}{(\frac{b^2}{ac}-b+ac})\cdot\log_{b}{(\frac{c^2}{ab}-c+ab})\cdot\log_{c}{(\frac{a^2}{bc}-a+bc})\geq1,
$$
where $a,b,c \in (0,1)$.
I tried to solve it in this way:
$$\log_{a}{(\frac{b^2}...
- 181