Joe has hooked up two function machines as shown in the diagram. When he puts $4$ into the “$3^{\text{input}}$” machine, $81$ comes out. Then the $81$ goes into the “$\log_{\text{base}3}$” machine.

1. What comes out of the “$\log_{\text{base}3}$” machine when $81$ goes in?

   Answer (a):

   $\log_381=4$ because $3^4=81$.

2. Now put $2$ into the “$3^{\text{input}}$” machine. What number comes out of this machine and goes into the “$\log_{\text{base}3}$” machine? What then comes out of the “$\log_{\text{base}3}$” machine?

   Answer (b):

   $3^{2} = 9$.

   $9$ goes into the $\log_3$ machine.

   $\log_39=2$ because $3^{2} = 9$.

3. If $x$ goes into the “$3^{\text{input}}$” machine, $y$ comes out. If $y$ then goes into the “$\log_{\text{base}3}$” machine, what comes out? If $y$ then, …

   Answer (c):

   $x$

4. If $x$ goes into the “$\log_{\text{base}3}$” machine, $y$ comes out. Write this in mathematical notation.

   Answer (d):

   $\log_3x=y$

5. If this same $y$ goes into the “$3^{\text{input}}$” machine, the original $x$ comes out. Write this in mathematical notation.

   Answer (e):

   $x = 3^{y}$

6. Do your answers to parts (d) and (e) say exactly the same thing?

   Answer (f):

   Yes