Solve-for-x-x-x-x-729-

Question Number 58899 by Tawa1 last updated on 01/May/19

S olve for x : x^{x^{x}} = 729

Commented by mr W last updated on 01/May/19

y o u {c a n}^{'} t s o l v e t h i s u s i n g L a m b e r t f u n c t i o n .

b u t y o u c a n s o l v e x^{x^{x^{x}}} = 729 .

Commented by Tawa1 last updated on 01/May/19

Ohh, why it cannot work sir, and please show me the x^{x^{x^{x}}} = 729

Commented by mr W last updated on 02/May/19

{(x^{x})}^{x^{x}} = a

l e t t = x^{x}

\Rightarrow t^{t} = a

\Rightarrow t = \frac{\ln a}{W (\ln a)}

\Rightarrow x^{x} = \frac{\ln a}{W (\ln a)} = b

\Rightarrow x = \frac{\ln b}{W (\ln b)} = \frac{\ln [\frac{\ln a}{W (\ln a)}]}{W {\ln [\frac{\ln a}{W (\ln a)}]}}

Commented by Tawa1 last updated on 01/May/19

God bless you sir .

Answered by MJS last updated on 01/May/19

x^{x^{x}} = x^{(x^{x})} = 729

approximation leads to

x \approx 2.364994

{(x^{x})}^{x} = x^{(x^{2})} = 729 \Rightarrow x \approx 2.617397

Commented by Tawa1 last updated on 01/May/19

But different answer sir . x^{x^{x}} \neq x^{x^{2}}

Commented by Tawa1 last updated on 01/May/19

And any workings sir

Commented by MJS last updated on 01/May/19

I know x^{x^{x}} \neq x^{x^{2}} but I {wasn}^{'} t 100 % sure if

you know this too because it {doesn}^{'} t seem

to be common knowledge .

I approximated with an electronic calculator

so I {can}^{'} t show any workings

I calculated f (x) = x^{x^{x}} - 729 starting with

f (2) = - 713

f (3) = 7.62 \times 10^{12}

f (2.5) = 7831.31 \dots

f (2.1) = - 695.08 \dots

f (2.2) = - 641.82 \dots

f (2.3) = - 442.76 \dots

f (2.4) = 554.40 \dots

f (2.35) = - 148.87 \dots

f (2.36) = - 54.00 \dots

f (2.37) = 59.19 \dots

f (2.365) = . 067773 \dots

f (2.364) = - 11.13 \dots

f (2.3645) = - 5.55 \dots

and so on

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