n*log(n,2) = 18*10^7

solve for n.

log(n,2) is equivalent to log(n) with base = 2.

Guest Jun 2, 2017

#1**0 **

This is how you should write it:

n*log(2, n) = 18*10^7

n≈ 7.8582779588575599198532262×10^6 [Still Newton-Raphson method-See your last post]

Guest Jun 2, 2017

edited by
Guest
Jun 2, 2017

#2**+2 **

You could also do this by simple iteration.

\(\text{First scale the problem to deal with more manageable numbers. Let }x=\frac{n}{18\times10^7}\\.\\\text{Then the equation becomes: }x\log_2(18\times10^7x)=1 \\.\\\text{Rewrite in the form: }x=\frac{1}{\log_2(18\times10^7x)} \text{ and use a simple iterative approach.}\\.\\ x_0=1\\ x_1=\frac{1}{\log_2(18\times10^7*x_0)}\rightarrow 0.036465\\ x_2=\frac{1}{\log_2(18\times10^7*x_1)}\rightarrow0.044158\\ ...\\ x_8=\frac{1}{\log_2(18\times10^7*x_7)}\rightarrow0.0436571\\.\\ n=0.0436571\times18\times10^7\approx7858278\)

Alan
Jun 3, 2017

#3**+1 **

Thanks very much Alan. Are you familiar with "Lambert W function?", sometimes called "Product log function?". Thank you again.

Guest Jun 3, 2017

#4**+2 **

Yes, I know about the LambertW function. This is usually written as w*e^w = x (where w is to be found given a value of x), or as w + ln(w) = ln(x). I didn't think to seek a solution to your problem in terms of LambertW as I suspect most people using this site are not familiar with it. (Also, not immediately obvious how to manipulate your equation to LambertW form!)

.

Alan
Jun 3, 2017