All Questions
11
questions
-2
votes
1
answer
54
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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
7
votes
1
answer
139
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Trying to find a NICE form of : $\sum_{m=1}^{n}\lfloor\log_2m\rfloor$ [ Mathematical Gazette 2002 ]
$Q.$ Find a NICE form of : $$\sum_{m=1}^{p}\lfloor\log_2m\rfloor$$
APPROACH : We have , $$\lfloor\log_21\rfloor⠀\color{red}{\lfloor\log_22\rfloor}⠀\lfloor\log_23\rfloor⠀\color{red}{\lfloor\log_24\...
- 688
3
votes
4
answers
119
views
For positive integer $n$, why is $\lfloor \log_{10}(2^n)\rfloor + \lfloor \log_{10}(5^n)\rfloor + 2 = n+1$?
For positive integer $n$, why is $\lfloor \log_{10}(2^n)\rfloor + \lfloor \log_{10}(5^n)\rfloor + 2 = n+1$?
This question comes from counting the number of digits of $10^n$ in terms of the number of ...
- 387
9
votes
1
answer
194
views
Solve $\lfloor \ln x \rfloor \gt \ln \lfloor x\rfloor$
The question requires finding all real values of $x$ for which $$\lfloor \ln x\rfloor \gt \ln\lfloor x\rfloor $$ To start off, one could note that $$\lfloor \ln x \rfloor =\begin{cases} 0,& x\in[1,...
- 14.5k
1
vote
1
answer
128
views
Equation with inverse trigonometric functions and logarithms
Solve over reals
$$\log_\frac{\pi}{2}\left(\arcsin\, \{x\}\right)+\log_\frac{\pi}{2}\left(\arccos\,\{x\}\right)=\frac{2}{\log_\frac{\pi}{4}\left(\arctan e^{\lfloor x\rfloor} + \operatorname{...
user750196
0
votes
0
answers
30
views
Using "ceil' To Make A function With seamless Changes
I'm trying to use ceil and or floor to accomplish something arrived at in another post. See lower two images for details: The uppermost image (of the two) is my question and the next is the answer.
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- 73
0
votes
0
answers
94
views
Proving $\lfloor e^x \rfloor =\lfloor e^{\lfloor x \rfloor} \rfloor$
Prove that $$\lfloor e^x \rfloor =\lfloor e^{\lfloor x \rfloor} \rfloor \tag{1}$$
I was actually trying to prove $$\lfloor \sqrt{x} \rfloor=\lfloor \sqrt{\lfloor x \rfloor} \rfloor$$ and i ...
- 10.2k
2
votes
1
answer
276
views
Are there ways to solve miscellaneous equations such as $\sin x=\log [x]$ without drawing the graphs?
Consider the example $$\sin x=\log [x]$$ where $[\,·\,]$ represents Greatest Integer Function. It is a miscellaneous equation, and I have been told that the only way to solve it is to draw the graphs ...
- 63
1
vote
1
answer
571
views
Simple proof that $\lfloor \log_2 k \rfloor + 1 = \lceil \log_2 (k + 1) \rceil$
Is there any way to show for $k \in \mathbb{N}$
$$\lfloor \log_2 k \rfloor + 1 = \lceil \log_2 (k + 1) \rceil$$
without casework, or of little of it as possible? I've tested it for some integers and ...
- 353
5
votes
5
answers
351
views
Solve floor equation over real numbers: $\lfloor x \rfloor + \lfloor -x \rfloor = \lfloor \log x \rfloor$
Consider : $\lfloor x \rfloor + \lfloor -x \rfloor = \lfloor \log x \rfloor$. How we can solve it over real numbers?
My try : I tried to solve it in several intervals but didn't get any result.
...
- 4,359
3
votes
2
answers
132
views
Calculation of all positive integer $x$ for which $\lfloor \log_{2}(x) \rfloor = \lfloor \log_{3}(x) \rfloor \;,$
Calculation of all positive integer $x$ for which $\displaystyle \lfloor \log_{2}(x) \rfloor = \lfloor \log_{3}(x) \rfloor \;,$
where $\lfloor x \rfloor $ represent floor function of $x$.
$\bf{My\;...
- 53.7k