WEI KIT RETURNS!

Wei Kit sure loves exponents! He's decided to rewrite all the numbers around him so that they have an exponent. For example, instead of writing the number $2$, he writes:

$3 ^ { \operatorname { log } _ { 3 } 2 }$

He insists he can also write the number $2$ as $5 ^ { \operatorname { log } _ { 5 } 2 }$ and $6 ^ { \operatorname { log } _ { 6 } 2 }$!

1. Explain why Wei Kit's expressions all equal $2$.

   Hint (a):

   Consider $3^{\operatorname{log}}{_3x}$
   First consider just the exponent: It can be translated as $'3$ to the power of something that makes $3$ become $x$.'
   Now consider the entire expression: $3$ is being raised to that 'something' that makes $3$ become $x$...
   Consequently, the expression $= x$.

2. Use Wei Kit's method to rewrite the number $9$ in two different ways.

   Answer (b):

   One way: $9 = 3^{\operatorname{log}}{_3}^9$
   Find another way.

3. Simplify this expression: $4 ^ { ( \operatorname { log } _ { 4 } 5 ) x }$

   Hint (c):

   Refer to previous hints.

4. How could Wei Kit rewrite $2^x$ so that it has a base of $7$?

   Hint (d):

   One way: $7 = 7^{??????}$