MHB Inequation with just 3 solutions

  • Thread starter Thread starter Vali
  • Start date Start date
AI Thread Summary
The discussion focuses on finding the number of natural number values "n" such that the inequality f(n/x) >= 1 has exactly three solutions in natural numbers. The function f is defined as f(x) = log_3(x), and the inequality simplifies to n >= 3x. Participants agree that for n = 3, the corresponding values of x must be limited to {1, 2, 3} to yield exactly three solutions. The correct answer to the problem is identified as 3. The conversation highlights the nuances of interpreting the inequality and the implications of the logarithmic function.
Vali
Messages
48
Reaction score
0
I have the following function:
f: (0,infinity) -> R
f(x)=log_3(x) (the base is 3)
I need to find the number of "n" values ( n is a natural number except 0 N*) such that this inequation: f(n/x) >= 1 to have just 3 solutions in N*.
A. infinity
B. 6
C. 9
D. 26
E. 3 (correct answer)
 
Mathematics news on Phys.org
Maybe I'm reading this wrong, but it looks to me like you need to count the number of natural numbers greater than \(3x\), which for any choice of \(x\) allowed, would be countably infinite.
 
Yes, I got the same result n>=3x
Maybe I wrote the sentence is a wrong way because I translated it from romanian.
I posted o picture below.
Exercise number 45.
The number of n values ( n natural ) for which the inequation f(n/x) >= 1 has exactly 3 solutions in N* is: ...
 

Attachments

  • 45.PNG
    45.PNG
    20.4 KB · Views: 94
Vali said:
I have the following function:
f: (0,infinity) -> R
f(x)=log_3(x) (the base is 3)
I need to find the number of "n" values ( n is a natural number except 0 N*) such that this inequation: f(n/x) >= 1 to have just 3 solutions in N*.
A. infinity
B. 6
C. 9
D. 26
E. 3 (correct answer)
This is a weird one. Here's my guess.
[math]log_3 \left ( \dfrac{n}{x} \right ) \geq 1[/math]

Since the log function is continuous we can take the exponent of base 3 on both sides:
[math]3^{ log_3 (n/x) } \geq 3^1[/math]

[math]\dfrac{n}{x} \geq 3[/math]

[math]n \geq 3x[/math]

So to have only 3 solutions, n = 3, gives us a domain for x as {1, 2, 3}.

-Dan
 
Thank you very much for the help :)
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Fermat's Last Theorem has long been one of the most famous mathematical problems, and is now one of the most famous theorems. It simply states that the equation $$ a^n+b^n=c^n $$ has no solutions with positive integers if ##n>2.## It was named after Pierre de Fermat (1607-1665). The problem itself stems from the book Arithmetica by Diophantus of Alexandria. It gained popularity because Fermat noted in his copy "Cubum autem in duos cubos, aut quadratoquadratum in duos quadratoquadratos, et...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...

Similar threads

Replies
13
Views
2K
Replies
2
Views
1K
Replies
1
Views
2K
Replies
4
Views
2K
Replies
1
Views
1K
Replies
7
Views
2K
Back
Top