x-x-100-find-x-

Question Number 16906 by tawa tawa last updated on 28/Jun/17

x^{x} = 100, find x .

Answered by mrW1 last updated on 28/Jun/17

x^{x} = 100

x = 100^{\frac{1}{x}}

x = e^{\frac{1}{x} \ln 100}

\frac{1}{x} e^{\frac{1}{x} \ln 100} = 1

(\frac{1}{x} \ln 100) e^{\frac{1}{x} \ln 100} = \ln 100

\Rightarrow \frac{1}{x} \ln 100 = W (\ln 100)

\Rightarrow x = \frac{\ln 100}{W (\ln 100)} \approx \frac{4.60517}{W (4.60517)}

\approx \frac{4.60517}{1.28018} = 3.59728

Generally :

solution for x^{x} = a is

x = \frac{\ln a}{W (\ln a)}

Commented by tawa tawa last updated on 28/Jun/17

God bless you sir . i really appreciate .