# How to solve this logarithmic equation?

I want to solve this equation: $$8n^2 = 64n\log_{\ 2}(n)$$

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!

• Sorry, numerical methods only. – Gerry Myerson Jun 29 '14 at 11:45

## 1 Answer

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: $$x_{i+1}=x_i-\frac{8log_2(n)-n}{\frac{1}{n*ln(2)}-1}$$ $$x_0=1.000$$ $$x_1=3.259$$ $$x_2=21.877$$ $$x_3=36.581$$ $$x_4=41.748$$ $$x_5=43.116$$ $$x_6=43.453$$ $$x_7=43.534$$ $$x_8=43.553$$ $$x_9=43.558$$ $$x_{10}=43.559$$ $$x_{11}=43.559$$ As you can see, the function takes a long time to converge, but it does converge nonetheless.

• How about the other solution, an approximation of which is $1.0999970302376094010$? – gniourf_gniourf Jun 29 '14 at 11:58