Natural Log Inequality: True or Misunderstanding?

Click For Summary

Discussion Overview

The discussion centers around the inequality $(\ln n)^a < n$ and whether it holds true for all values of $a$, particularly in the context of $n$ approaching infinity. Participants explore the implications of this inequality and its validity.

Discussion Character

  • Debate/contested, Conceptual clarification, Mathematical reasoning

Main Points Raised

  • One participant claims that the inequality is false, using the example of $\ln^3(e^2)$ and noting that $e^2 \approx 7.4$.
  • Another participant reiterates the same example to support the claim that the inequality does not hold.
  • A later post emphasizes that the context of the discussion is important, specifically mentioning that the inequality should be considered as $n$ approaches infinity.
  • One participant challenges the initial claim by pointing out that the original statement was made without considering the limit context.
  • A participant suggests that the question relates to limits and proposes moving the discussion to a Pre-Calculus context.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the validity of the inequality. There are competing views regarding its truth, particularly in the context of limits as $n$ approaches infinity.

Contextual Notes

The discussion highlights the importance of context, specifically the behavior of the inequality as $n$ approaches infinity, which some participants believe was overlooked in earlier statements.

tmt1
Messages
230
Reaction score
0
I was talking to my professor and she said that $(ln n)^a < n$ for all values of $a$. Is this true or was I misunderstanding?
 
Physics news on Phys.org
It's false. Consider $\ln^3\left(e^2\right)$ and note that $e^2\approx7.4$.
 
greg1313 said:
It's false. Consider $\ln^3\left(e^2\right)$ and note that $e^2\approx7.4$.

I forgot an important detail. This is in context of $n$ approaching infinity.
 
You 'forgot' that? You completely denied it when you said "for all n"!
 
This is essentially a question of limits - moved to Pre-Calculus.

tmt, do you think it's true? False? Explain your reasoning.
 

Similar threads

High School Why is natural log abbreviated as ln and not nl ?
  • · Replies 8 ·
Replies
8
Views
17K
High School Can a log have multiple bases?
  • · Replies 44 ·
2
Replies
44
Views
5K
MHB S6.7.1.13 natural log Integration
  • · Replies 3 ·
Replies
3
Views
2K
MHB What is the difference between ln 4 and log 4?
  • · Replies 3 ·
Replies
3
Views
2K
MHB Understanding the Inequality for Solving Limits with Exponential Terms
  • · Replies 4 ·
Replies
4
Views
1K
Undergrad Why is ln(x) Important in Mathematical Formulas?
  • · Replies 7 ·
Replies
7
Views
3K
MHB Natural logs solve ln⁡((x-1)/(x-3))=2
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad I found a *useful* method to calculate log(a+b), check it out
  • · Replies 8 ·
Replies
8
Views
3K
High School How to express ##3^{\sqrt(2)}## in terms of natural logarithms
  • · Replies 10 ·
Replies
10
Views
2K
Undergrad Integral involving square and log
  • · Replies 4 ·
Replies
4
Views
2K