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:

$\large 3^{\log_{3}(2)}$

He insists he can also write the number $2$ as $5^{\log_{5}(2)}$ and $6^{\log_{6}(2)}$!

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

   Hint (a):

   Consider: $3^{\log_{3}(x)}$

   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: $9=3^{\log_{3}(9)}$
   Find another way.

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

   Hint (c):

   Refer to previous hints.

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

   Hint (d):

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