How to solve this logarithmic equation? I want to solve this equation: 8n^2=64nlog _2(n) After some steps, I get to a point in which I believe, the only way to proceed is to apply something like Bolzano's or Newton's method to find a solution.

Daniella Reyes

Daniella Reyes

Answered question

2022-09-23

How to solve this logarithmic equation?
I want to solve this equation:
8 n 2 = 64 n log   2 ( n )
After some steps, I get to a point in which I believe, the only way to proceed is to apply something like Bolzano's or Newton's method to find a solution.
I get to: n = 8 log   2 ( n )
Of course with big numbers applying Bolzano would be very tedious and this is why I want to ask you if there is an analytical way of solving this, not by approximations.
Thanks a lot!

Answer & Explanation

Micah Hobbs

Micah Hobbs

Beginner2022-09-24Added 8 answers

No analytical solution exists in terms of elementary functions. However, simplifying the expression is advisable before running Newton's Method, if it is practical to simplify. Solution would be:
x i + 1 = x i 8 l o g 2 ( n ) n 1 n l n ( 2 ) 1
x 0 = 1.000
x 1 = 3.259
x 2 = 21.877
x 3 = 36.581
x 4 = 41.748
x 5 = 43.116
x 6 = 43.453
x 7 = 43.534
x 8 = 43.553
x 9 = 43.558
x 10 = 43.559
x 11 = 43.559
As you can see, the function takes a long time to converge, but it does converge nonetheless.

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