MHB Equality of natural ln function

AI Thread Summary
The discussion centers on the inequality \( (\ln(n))^4 < n^{1/4} \) for \( n > 1 \), which is claimed to hold only for \( n = 1 \) and \( n = 2 \) when \( n \) is a natural number. Participants clarify that the statement is an inequality rather than an equality. The possibility of generalizing this inequality to \( (\ln(n))^b < n^{1/b} \) for \( n > 1 \) is questioned, suggesting that the original inequality may not be valid for all \( n \). The conversation emphasizes the need for careful consideration of the conditions under which such inequalities hold true. The discussion concludes with a focus on the limitations of the proposed generalization.
tmt1
Messages
230
Reaction score
0
I have this equality:

$$ (\ln\left({n}\right))^4 < {n}^{\frac{1}{4}} $$ where $ n > 1$

Can I derive a law from this such that

$$ (\ln\left({n}\right))^b < {n}^{\frac{1}{b}} $$ where $n > 1$ ?
 
Mathematics news on Phys.org
Hi tmt. $\ln^4(n)<n^{1/4}$ is only true for $n\in\{1,2\}$, if $n\in\mathbb{N}$.

By the way, it's an inequality.
 
Last edited:

Thread 'Non-orthogonal bases'

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...
View full post »

Thread 'Fermat's Last Theorem'

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...
View full post »

Thread 'What Exactly is Dirac’s Delta Function? - Insight'

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...
View full post »

Similar threads

I ##1^x =2## has complex solutions?
Replies
7
Views
2K
I Am I misapplying something here? (Exponentials; Euler's identity)
Replies
3
Views
191
MHB Seth's question via email about a Laplace Transform
Replies
1
Views
10K
MHB How Can You Rewrite an Equation to Express y as a Function of x?
Replies
3
Views
1K
A How to sum an infinite convergent series that has a term from the end
Replies
15
Views
2K
MHB Can you find $f^{(n)}(1)$ with the given conditions?
Replies
2
Views
1K
B How does the Riemann Hypothesis/Riemann Zeta function even work?
Replies
17
Views
911
I Relationship between factorials and squares of natural numbers
Replies
12
Views
3K
A Are these two optimization problems equivalent?
Replies
1
Views
1K
I Solve ln(a/b+1) Using Identity ln(a+b)=ln b + ln(a/b+1)
Replies
10
Views
3K

Hot Threads

Recent Insights

Back
Top