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

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SUMMARY

The equation 8n² = 64n lg(n) is a transcendental equation that typically does not have an algebraic solution. Participants in the discussion suggest using graphical methods or numerical approximation techniques, such as trial and error, to find solutions. One participant reported using Mathematica and found an approximate solution of n ~ 6.5. The inequality 8n² < 64n lg(n) is also explored, with a focus on integer values of n as they pertain to algorithm input sizes.

PREREQUISITES
  • Understanding of transcendental equations
  • Familiarity with logarithmic functions, specifically lg(n)
  • Basic knowledge of numerical approximation techniques
  • Experience with mathematical software like Mathematica
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  • Research graphical methods for solving transcendental equations
  • Explore numerical approximation techniques for finding roots
  • Learn about the properties of logarithmic functions in algorithm analysis
  • Investigate the use of Mathematica for solving complex equations
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Mathematicians, computer scientists, and students studying algorithm analysis who need to solve complex equations and understand their implications in computational contexts.

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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