Real solution of equations of 2018

Question

Question:

What can we say about positive real solutions to the equations

$x^{x^{2018}}=2018$ and $x^{x^{x^{... }}}=2018$

Options:

1)One exists,other does not (Correct answer)

2)they exist and are same(Wrong)

3)they exist but are different.(Wrong)

My approach :

The second infinite power sequence is either 1 or diverges, therefore there is no solution.

Conclusion:

*Any solution or guidance on how to approach and solve this problem would enable me to increase my knowledge and others who come across this question *

Answer 1

$$ x^{x^{2018}} = 2018 \implies x^{2018 x^{2018}} = 2018^{2018} \\ \implies (x^{2018})^{( x^{2018})} = 2018^{2018} \\ \text{this allows at least one possible solution} \\ \implies (x^{2018}) = 2018 \\ x = 2018^{1/2018} \approx 1.003778111307819 $$

Answer 2

As noted in my comments above, there is no value of $x$ such that the infinite "power tower" converges to 2018. As to the other problem, taking logarithms twice gives $$\log \log x + 2018 \log x = \log \log 2018.$$

When $x=1$ the left-hand side is $-\infty$ and when $x \rightarrow \infty,$ the left hand side goes to $\infty,$ so by the intermediate value theorem, there is a solution. If fact, since the left-hand side is clearly an increasing function of $x$, the solution is unique. (Approximately $1.003777,$ but that's pretty rough.)