Solving the Equation: 8n^2 = 64 n lg(n) with Step-by-Step Guide

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Discussion Overview

The discussion centers around solving the equation 8n^2 = 64 n lg(n) for n, with a focus on both integer and real solutions. Participants explore methods for isolating n and consider the implications of the equation in the context of algorithm running times.

Discussion Character

  • Mathematical reasoning, Homework-related, Debate/contested

Main Points Raised

  • One participant asks how to isolate n in the equation 8n^2 = 64 n lg(n).
  • Another participant suggests that the equation is typically transcendental and recommends graphical methods or trial-and-error approaches for finding solutions.
  • A different participant mentions having used Mathematica to find an approximate solution of n ~ 6.5 but inquires about the possibility of an algebraic solution.
  • One participant raises a question about whether n is supposed to be an integer.
  • A later reply confirms that n is an integer, as it represents input size for algorithms, but expresses interest in both integer and real cases.

Areas of Agreement / Disagreement

Participants express differing views on the solvability of the equation, with some suggesting numerical methods while others seek analytical approaches. There is no consensus on whether an algebraic solution exists.

Contextual Notes

Participants note that the equation may have zero, one, or a maximum of two solutions, and the discussion includes considerations of the equation's implications for algorithm running times.

Who May Find This Useful

This discussion may be useful for students studying algorithm analysis, those interested in solving transcendental equations, or anyone exploring numerical methods for finding solutions to mathematical problems.

Niels
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How do I solve the following equation?

8n^2 = 64 n lg(n); (0 < n)

n = 8lg(n)
10^n = 10^8 n
...? How do I isolate n?
 
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Niels said:
How do I solve the following equation?
8n^2 = 64 n lg(n); (0 < n)
n = 8lg(n)
10^n = 10^8 n
...? How do I isolate n?

Who says u can?It's typically a transcendental equation.I suggest either graphical method (intersaction of graphs) (done by computer,maybe),or taking a calculator and "solving it through tries".Your equation may have 0,1 or maximum 2 solutions.

Daniel.
 
I already did that with mathematica and got that one solution is x ~ 6.5... I just wanted to know if there was any algebraic solution...

I study running times of some algorithms and got this questions: for that values of n is the following inequality true:
8n^2 < 64 n lg(n)

Is there no analytical approach. This is a potential exam question and were not allowed to use calculators...
 
Is n supposed to be an integer?
 
Yes, n is integer (input size for algorithm) but I'm interested in both cases. (real/integer)
 

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