Solving logarithmic equation for n I have the following equation and I am trying to isolate n: 8n^2 = 64 n log_2 n Haven't done algebra in years and can't figure out how to get rid of the log_2

Simone Watts

Simone Watts

Answered question

2022-11-05

Solving logarithmic equation for n
8 n 2 = 64 n log 2 n
Haven't done algebra in years and can't figure out how to get rid of the log 2 .

Answer & Explanation

Jackson Trevino

Jackson Trevino

Beginner2022-11-06Added 14 answers

As said in comments, because of the logarithm, n > 0; so, as you wrote, the equation simplifies to
n = 8 n log 2 n
which does not have a solution in terms of elementary functions.
However, any equation which can be written as
a + b x + c log ( d + e x ) = 0
has a solution which is expressed in terms of the Lambert function. In your case, there are two solutions given by
n 1 = 8 log ( 2 ) W ( log ( 2 ) 8 ) 1.099997030
n 2 = 8 log ( 2 ) W 1 ( log ( 2 ) 8 ) 43.55926044
If you do not want (or cannot) to use Lambert function, you could use a root-finding method such as Newton. Starting with a "reasonable" guess n 0 , this will find the solution of
f ( n ) = n 8 n log 2 n = 0
updating the guess according to
n k + 1 = n k f ( n k ) f ( n k )
For your case, the iterative schme will then be
n k + 1 = 8 n k ( log ( n k ) 1 ) n k log ( 2 ) 8
Let us start with a very poor estimate such as n 0 = 20; Newton successive iterates will then be 54.4636, 43.8990, 43.5597, 43.5593 which is the solution for six significant figures.

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