How can I write the numbers 5 and 7 as some sequence of operations on three 9s? I want to make the numbers $1, 2, ..., 9$ using exactly three copies of the number $9$ and the following actions: addition, subtraction, multiplication, division, squaring, taking square roots, and other action.
How can we make the numbers $5$ or $7$?
For example, we can make the below numbers using exactly three copies of the number 9.


*

*$1=\dfrac{\sqrt 9\times\sqrt9}{9}$

*$2=\dfrac{9+9}{9}$

*$3=\dfrac{\sqrt9\times9}{9}$

*$4=\dfrac{9}{9}+\sqrt9$

*$5=\,?$

*$6=\dfrac{9+9}{\sqrt9}$

*$7=\,?$

*$8=9-\dfrac{9}{9}$

*$9=9+9-9.$


Now, how can we make the numbers 5 and 7?

$5 = \dfrac{\ln(9 \times \sqrt{\sqrt{9}})}{\ln(\sqrt{\sqrt{9}})}$
$7 = \dfrac{\ln\left(\dfrac{9}{\sqrt{\sqrt{\sqrt{9}}}}\right)}{\ln\left(\sqrt{\sqrt{\sqrt{9}}}\right)}$
