Thread: help for my son

1. help for my son

Can someone help me solve this for my son? I have gone blank!

I need to solve this for Y.

C.X^Y = P.Y

thanks

2. Re: help for my son

As far as I know, Y(X) cannot be expressed in elementary functions in general.

3. Re: help for my son

Mark.. Emakarov is right

you cannot express y as an algebraic expression ...since it is trancedental.
such type of equations can be solved only by approximation or graphically .

4. Re: help for my son

thanks guys. I thought I was losing it, but what you say makes sense.

5. Re: help for my son

$\displaystyle X^y= e^{ln(X^Y)= e^{Y ln(X)}$ so the equation can be written as $\displaystyle Ce^{Y ln(X)}= PY$ or $\displaystyle \frac{C}{P}= Ye^{-Yln(X)}$. Let Z= -Yln(X). Then $\displaystyle Y= -Zln(X)$ and $\displaystyle e^{-Yln(X)}= e^Z$ so the equation becomes $\displaystyle \frac{C}{P}= -Zln(X)e^Z$ and then $\displaystyle -\frac{C}{PZln(X)}= Ze^Z$.

Now apply "Lambert's W function" to both sides. "Lambert's W function" is defined as the inverse function to $\displaystyle f(x)= xe^x$ so $\displaystyle W(Ze^Z)= Z= W\left(-\frac{C}{PZln(X)}\right)$. Of course, "Lambert's W function" is NOT an "elementary function" so emakarov is completely correct.