How can two different utility functions represent the same preferences?

Question

I have this question for microecon that asks do the following utility functions represent the same preferences:

1. $u(x_1, x_2) = x1 \cdot x2, \; v(x_1, x_2) = \ln x_1 + \ln x_2$

2. $u(x_1, x_2) = x1 \cdot x2, \; v(x_1, x_2) = x_1 + x_2$

How do I go about approaching this problem since they are two different functions?

Answer 1

First of all, an utility function $u: X \rightarrow \mathbb{R}$ represents the preference relation $\succsim$ if: $$\forall a, b \in X, \; u(a) \geq u(b) \iff a \succsim b.$$

Well, if another function $v: X \rightarrow \mathbb{R}$ represents $\succsim$, then: $$\forall a, b \in X, \; v(a) \geq v(b) \iff a \succsim b.$$

But the above implies the following:

$$\forall a, b \in X, \; u(a) \geq u(b) \iff v(a) \geq v(b).$$

Which means that both functions preserves the order relation, just as Herr K. said it. Another important part of Wikipedia's article is this one. You see, any monotonically increasing (and it must be increasing, not nondecreasing) function preserves the preference relation.

So now I think you will be able to answer both questions. As a tip for the first one, think about the $\ln(x)$ function: is it a monotonic increasing function?