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

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The equation 8n^2 = 64n lg(n) is identified as a transcendental equation, making it challenging to isolate n algebraically. Graphical methods or numerical approaches, such as trial and error, are suggested for finding solutions, with one known solution around n ~ 6.5. The discussion also raises the question of whether the inequality 8n^2 < 64n lg(n) holds for certain values of n, particularly in the context of algorithm running times. While n is specified as an integer, the interest extends to both real and integer solutions. The need for an analytical approach is emphasized due to exam constraints prohibiting calculator use.
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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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