# How do you solve (x-1)^[log(x-1)]=100(x-1)?

Jun 18, 2016

$x = 1 + {e}^{\frac{1}{2} \left(1 + \sqrt{1 + 4 L o {g}_{e} \left(100\right)}\right)}$

#### Explanation:

Making $y = x - 1$ we have

${y}^{{\log}_{e} y} = 100 y \to {y}^{{\log}_{e} y - 1} = 100$

Applying $\log$ to both sides

$\left({\log}_{e} y - 1\right) {\log}_{e} y = {\log}_{e} 100$

Solving for ${\log}_{e} y$ we have

${\log}_{e} y = \frac{1}{2} \left(1 \pm \sqrt{1 + 4 L o {g}_{e} \left(100\right)}\right)$

so

$Y = x - 1 = {e}^{\frac{1}{2} \left(1 \pm \sqrt{1 + 4 L o {g}_{e} \left(100\right)}\right)}$

Finally considering that $x - 1 > 0$

$x = 1 + {e}^{\frac{1}{2} \left(1 + \sqrt{1 + 4 L o {g}_{e} \left(100\right)}\right)}$