Inequality Problem from Spivak's Calculus: Chapter 1, Problem 4, Subproblem XI

[Why?]

Homework Statement

Find all numbers x for which:

2^x<8

Homework Equations

The Attempt at a Solution

I really haven't been able to figure this one out.

 

[Axiom17]

Do you have any calculations to show?

I assume it means integer values? If so surely isn't it just:

$$x=-\infty, ..., 0, \pm 1, \pm 2, \pm 3$$

Unless I'm missing the point somewhere? It seems a bit simple though.

 

[hunt_mat]

Note that $$8=2^{3}$$, 2^{x} in monotonically increasing. So the question is, what values of x satisfy
$$2^{x}<2^{3}$$
can you say what values satisfy this equation?

 

[tmccullough]

Stated in a different way, $\log_2(t)$ is an increasing function. Inequalities remain true if you apply an increasing function.

 

[Why?]

Sorry I wasn't very clear.

Just thinking it through I know that 2³ is 8, so x<3.

However my difficulty was in proving it, using the mathematical context that Spivak uses.

hunt_mat's example makes a lot of sense to me and fulfills that need to explain it more concretely.

Thanks for your help!