How Can I Solve the Equation \(2^x = x^{10}\)?

[Dell]

2^x=x^10

ln(2^x)=ln(x^10)
x*ln(2)=10*ln(x)
ln(2)/10=ln(x)/x

from here how can i solve this??

i seem to keep on going in circles

 

[Bohrok]

You won't be able to find an explicit or exact answer for this kind of problem.

 

[Dell]

how can i go about it then?

 

[LCKurtz]

Numerically. Maple gives 1.077550150.

 

[Dell]

...n

 

[Dell]

i looked the Lambert W function up, and using it i get,

2^x=x¹⁰
(2^1/10)^x=x
1=x/(2^1/10)^x =====> (2^1/10)=a
1=x/a^x
1=x*e^-x*ln(a)
-ln(a)=[(-ln(a)*x)*e^(-ln(a)*x)

after applying W to each side

-ln(a)*x=W(-ln(a))
x=W(-ln(a)/-ln(a))

is this correct, and also how do i get a numerical answer for this??

 

[LCKurtz]

You might as well have solved the original equation numerically.

 

[Dell]

how would i have done that

 

[LCKurtz]

Newton's method for one.