My question is- $$(4x)^{{{{\sqrt x}^{\sqrt x}}^ \cdots}^\infty}=0.0625$$ How to solve it?
Options- (A)$2^{1/24}$ (B)$2^{1/48}$ (C)$4^{1/48}$ (D)$2^{1/96}$
I am confused how to solve this infinite power tree. Any help will be appreciated Note: Multiple options or none of the options may be correct My approach- Took log on both sides then simplified to $xlog4x=log5-3$, then punched out it on a calculator but it says that no real solutions exists.
(Too long for a comment.)
$(4x)^{{{{\sqrt x}^{\sqrt x}}^ \cdots}^\infty}=0.0625 \Longrightarrow (2\sqrt{x})^{{{{\sqrt x}^{\sqrt x}}^ \cdots}^\infty}=0.25$
Let $\sqrt{x}^{{\sqrt{x}}^{{\cdots \infty}}}=a$
We have $2^a\cdot a=0.25$
Solve for $a$ then solve for $x$?