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Problem - Pangkat Bertingkat tiga | Catatan Nurkholis

Problem – Pangkat Bertingkat tiga

Problem:

Diketahui bahwa $x$ merupakan bilangan real yang memenuhi:
$$x^{x^{x^{2013}}}=2013$$
Tentukan nilai $x$

Solusi:
Misalkan $y=x^{x^{2013}}$, maka $x^y=2013$ atau $x=2013^{1/y}$. Substitusikan kembali ke $y$ didapat:
$$

begin{aligned}
y&=x^{x^{2013}}\
y&=left(2013^{1/y}right)^{{left(2013^{1/y}right)}^{2013}}\
y&=left(2013right)^{left(frac{2013}{y}right)^{left(frac{2013}{y}right)}}\
y^{left(frac{y}{2013}right)}&=2013^{left(frac{2013}{y}right)}\
y^{y^y}&=2013^{2013^{2013}}\
y&=2013
end{aligned}

$$
Dengan demikian, $displaystyle x=2013^{frac{1}{2013}}$

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