Problem: If xxx⋅⋅⋅=2x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} =2xxx⋅⋅⋅=2, then xxx is equal to.
Answer Choices:
A. infinity B. 222 C. 24\sqrt[4]{2}42 D. 2\sqrt{2}2 E. none of these answers.
Solution:
If xxx⋅⋅⋅=2x^{x^{x^{\cdot^{\cdot^{\cdot}}}}} =2xxx⋅⋅⋅=2, then the exponent, which is again xxx⋅⋅⋅x^{x^{x^{\cdot^{\cdot^{\cdot}}}}}xxx⋅⋅⋅ is also 222, and we have x2=2x^{2}=2x2=2. Therefore x=2x=\boxed{\sqrt{2}}x=2.