# Question 6c76f

Nov 8, 2016

${y}^{\frac{1}{y}} = {\left(\frac{z}{x}\right)}^{\frac{1}{x}}$

#### Explanation:

$\frac{z}{x} = {y}^{\frac{x}{y}}$ applying the $\log$ function

$\log \left(\frac{z}{x}\right) = \frac{x}{y} \log y$ or

$\frac{1}{x} \log \left(\frac{z}{x}\right) = \frac{1}{y} \log y$ and finally

${\left(\frac{z}{x}\right)}^{\frac{1}{x}} = {y}^{\frac{1}{y}}$ this is the best I can do.

Nov 8, 2016

y=1/(e^(W(1/xln(x/z)))

where W is the Lambert function (on Wikipedia)

#### Explanation:

Let $h = \frac{1}{y}$

Then $\frac{z}{x} = \frac{1}{{h}^{x h}} \setminus \setminus \setminus \setminus$ so $\setminus \setminus \setminus {h}^{x h} = \frac{x}{z} \setminus \setminus \setminus$ and $\setminus \setminus \setminus {h}^{h} = {\left(\frac{x}{z}\right)}^{\frac{1}{x}}$

$h \ln h = \frac{1}{x} \ln \left(\frac{x}{z}\right)$

Using the example 5 in Wikipedia

h=e^(W(1/xln(x/z))

but $\setminus \setminus \setminus h = \frac{1}{y} \setminus \setminus \setminus$ so

y=1/(e^(W(1/xln(x/z)))#