Yes, the graphs seem to be visual proof. And my rather wordy "proof" above was essentially saying what the graph would say. But that feels a little circumstantial to me and may or may not be solid enough proof to include it as a rather pivotal step in a larger proof (the crux of number theory being to avoid any ambiguity where you just never know, somewhere waaaaay out on the number line you might run across a weird hiccough in the procedings).

A little bit of hand waving is acceptable in proofs. You don't have to go back and prove that 2>1 every time you start a proof. And I know this one is pretty trivial. I'm just not well versed in the theory enough to know if it's trivial enough for me to get away with leaning on some higher level concepts (exponential and linear growth) rather than formally proving the nuts and bolts.

Yes, it is. A very striaght forward one, but a provable theory none the less.

Like I said, I'm aware that it's a fairly self-evident truth. But in terms of mathematic rigor, would more detail be required (the answer may very well be no)?

Is it warm in here, or is it just me... anyone got a thermometer? ;)

Mori's getting all moist in the math thread!

</invasion of math thread by non-geek>

Solving for y1=x*7 at x=3; y=21.
Solving for y2=10^(x-1) at x=3; y=100.

Slope:
Derivative of y1: y1'=7 at x=3 (and all values of x)
Derivative of y2: y2'=10^(x-1); at x=3, y2'=100
Second derivative of y2: y2''=10^(x-1)

y1<y2 at x=3
y1'<y2' at x=3
y2'' is positive for all values of x indicating that y2' increases as x increases
therefore y1<y2 for all integers where x>2

That's way more complicated than just a pretty picture showing the plot of each equation on one graph, but I think it satisfies the burden of proof.

And also a slightly more formal way of saying "linear growth vs. exponential growth," got it. I think that's good enough for me.

But where do you get y2'=y2? Isn't the derivative of 10^x=10^x ln (10)? (too brain fried to figure out how the "x-1" affects that)

I have to ask.

Where in real life did this come from?

Nevermind, ln not log.

My warped mind and a cold that kept me awake.

A puzzle from NPR's Sunday Puzzle got me off on a tangent.

Good grief.

If you're that bored I have some math homework you can do ;)