  ### Home > CALC > Chapter 5 > Lesson 5.3.3 > Problem5-131

5-131.

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 $2$.

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.

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

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

Refer to previous hints.

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

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