X^2*log(2)x=256^2 (2) refers to base 2Is ther any rule to solve

[kind]

x^2*log(2)x=256^2

(2) refers to base 2
Is ther any rule to solve for x.

 

[arildno]

No, there isn't any general rule to get an exact solution to that equation.

 

[Char. Limit]

x can be solved for here, but it involves a function that you've probably not heard of, called the "Product Log function", or the "Lambert-W function". It's defined as the inverse function of the function y=x*e^(x). In other words, if x=y*e^(y), then y=W(x).

This is the only way to solve for x in your function, by using Lambert's W Function.

EDIT: Or, if c=1, then x=0.

 

[jackmell]

x^2*log(2)x=256^2

(2) refers to base 2
Is ther any rule to solve for x.

Here's how to do it via the Lambert W function:

$$\frac{x^2}{\log(2)}\log(x)=(256)^2$$

$$\log(x)=\frac{\log(2)(256)^2}{x^2}$$

$$x=\exp\left\{\frac{\log(2)(256)^2}{x^2}\right\}$$

Now square both sides and divide by x^2:

$$1=\frac{1}{x^2} \exp\left\{\frac{2\log(2)(256)^2}{x^2}\right\}$$

$$2\log(2)(256)^2=\frac{2\log(2)(256)^2}{x^2}\exp\left\{\frac{2\log(2)(256)^2}{x^2}\right\}$$

and that's where we take the W-function of both sides:

$$\frac{2\log(2)(256)^2}{x^2}=W\left[2\log(2)(256)^2\right]$$

$$x=\pm \sqrt{\frac{2\log(2)(256)^2}{W\left[2\log(2)(256)^2\right]}}$$

Nothin' wrong with the W function is it guys?

 

[arildno]

It is not an elementary function.

To just about any equation you might define a function by which the solution can be represented.

So what?

That Lambert functions, Hankel functions, Bessel functions and whatnot are useful is not the issue here.

 

[jackmell]

It is not an elementary function.

To just about any equation you might define a function by which the solution can be represented.

So what?

That Lambert functions, Hankel functions, Bessel functions and whatnot are useful is not the issue here.

Come on Arildno. You made a boo-boo. Just fess-up. He asked how to solve it without qualification and what I showed is conceptually no different than using sines and cosines to solve an equation.

Stop special function discrimination: Equal rights for special functions.

 

[arildno]

Stop special function discrimination: Equal rights for special functions.

Nope.