#### Answer

For $9^x=20$: $1\lt x \lt2$
For $9^x=100$: $2\lt x \lt3$

#### Work Step by Step

We look at powers of $9$:
$9^{1}=9$
$9^{2}=81$
$9^{3}=729$
We know that the solution to $9^{x}=20$ must have an $x$ value between $1$ and $2$ because $20$ is between $9$ and $81$. We also know that the solution to $9^x=100$ must have an $x$ value between $2$ and $3$ because $100$ is between $81$ and $729$.