20點!數學超勁難題2^x=x^2數學高手入

這題最好就要試下使用一個很好的工具：Lambert W function（http://en.wikipedia.org/wiki/Lambert_W_function）。因為它是一個很罕有的用Lagrange inversion theorem（http://en.wikipedia.org/wiki/Lagrange_inversion_theorem#Lambert_W_function）能找到美妙的series form的function，這比找數值解好n倍。

2010-11-13 06:40:35 補充：

圖片參考：http://i212.photobucket.com/albums/cc82/doraemonpaul/yahoo_knowledge/crazyequ/crazyequ9.jpg

參考資料：
my wisdom of maths + http://www.wolframalpha.com/input/?i=2%5Ex%3Dx%5E2 + http://en.wikipedia.org/wiki/Lambert_W_function#Special_values + http://en.wikipedia.org/wiki/Lambert_W_function#Taylor_series + http://www.wolframalpha.com/input/?i=sum%28-1%29%5Ek%28ln2%29%5E%28k-1%29k%5E%28k-1%29%2F%282%5E%28k-1%29k%21%29%2Ck%3D1+to+inf

2^x=x^2
x‧log2=2‧logx
compare the two terms separately, we have
x=2 and log2=log x
so x=2 is one solution

2^x=x^2
take square on both sides
2^2x=x^4
4^x=x^4
same proof as above
x‧log4=4‧logx
compare the two terms separately, we have
x=4 and log4=log x
so x=4 is another solution

圖解法 :

http://www.wolframalpha.com/input/?i=+2%5Ex+-x%5E2%3D0

re:自由自在 ( 知識長 ) x = -0.766664695962123 點解出黎?
re:chung,我識微分,D完之後只計到x=2.牛頓堪根法計咩黎?

2010-11-07 01:49:08 補充：
咁用得圖解法既話,要點做?唔該講解下.如何用牛頓堪根法應用在這題中.

你識唔識計微分???如果識咁你可以用牛頓堪根法計..

2010-11-06 23:13:46 補充：
對唔住...諗諗下,如果唔用得圖解,應該都唔用得牛頓堪根法,因為牛頓堪根法都唔系準確值,只會隨次數的重複而增加準確值,但唔可以計出0誤差的值....
http://episte.math.ntu.edu.tw/java/jav_newton/index.html
呢個網有講....或者就算牛頓堪根法計唔到準確值,但可以學多d野~~

x = -0.766664695962123

2^x (not=) x^2

我想講呢到係log唔到2架wor...

When x = 2,
2^(2) = (2)^2
4 = 4
根本就log唔到.