Stuck.... urgent help needed!

[loststudent]

I was in the middle of a finding a minimum, when I got stuck with this frustrating :shock: equation. I need urgent help as I need to submit my paper by tomorrow:


x^(1/2)+ln(x^(1/2))=u-ln(d) [u and d are parameters).

Is there anything to be done with it to simplify the equation? In part two of the question, I am to assume that u=1 and d=2, but I still can't solve it.

Thanks!
:shock:

 

[mmm4444bot]

Combining radical and logarithmic terms often creates an equation that cannot be solved algebraically. I believe your equation is such because MVR5 software gives the following exact solution for x, in terms of the parameters u and d, as well as base e, using a specialized function named LambertW().

\(\displaystyle \sqrt{x} \;=\; LambertW\left ( \frac{e^u}{d} \right )\)

There are different ways to find an approximation for x, which happens to be just under 1/2, with the given parameter values.

Are you supposed to estimate the solution above using a graphical method ?

Or, perhaps, your course uses software to estimate the solution ?

Graphing calculator ?

Can you provide more detail on exactly what you're trying to do ?

 

[loststudent]

Thanks!

I was supposed to show the solution for x, but I get from your answer that there is no way to simplify the equation any further...
In the second part, I was asked to solve, when u=1 and d=2 - I am unable to do it on my own. I guess I should triple check my equations, maybe I got something wrong on the way..

Thanks again

 

[mmm4444bot]

I was asked to solve, when u = 1 and d = 2

You can get an estimate for the solution graphically. If you use software, you can get as much precision as you like.

Hold on ... dogs going berserk ...

 

[mmm4444bot]

Okay, we have u - ln(d) when u = 1 and d = 2, so that's easy to approximate: 0.30685

The equation is: sqrt(x) + ln(sqrt[x]) = 0.30685

Subtract ln(sqrt[x]) from both sides.

sqrt(x) = 0.30685 - ln(sqrt[x])

We can think of each side as its own function:

f(x) = x^(1/2)

g(x) = 1 - ln[2x^(1/2)]

I got the definition for g(x) by using 1 - ln(2) - ln(x^[1/2]), and applying properties, to simplify.

Plot functions f and g on the same coordinate system, and look for the intersection point of their graphs.

 

[loststudent]

Awesome, thanks!