natural log function

mikau · 2007-03-18 09:54:58

I was just thinking about a creepy property of logarithmic functions.

We know that if we have the graph of any function f(x), the graph of f(2x) is exactly the same except it is horizontally compressed. Half as wide.

but if f(x) happens to be ln(x), then ln(2x) is NOT horizontally compressed. Instead it is verticallly shifted by +ln(2) units. Does it break the rule? Nope. Compress the graph of ln(2x) untill its half as wide, and you end up with the exact same curve, only its ln(2) units higher.

Likewise, with a function like ln(x^2) you'd think the graph would accelerate at a much faster rate than the graph of ln(x), since the argument is accelerating instead of moving at a constant rate. Instead, it just moves at a constant 2 times as fast. Creepy or what?

It seems everything related to e is surrounded by unique, amazing properties!

Last edited by mikau (2007-03-18 09:57:55)

Sekky · 2007-03-18 11:16:04

because the exponential function acts as an isomorphism between additive and multiplicative groups

JaneFairfax · 2007-03-18 11:25:58

mikau wrote:

but if f(x) happens to be ln(x), then ln(2x) is NOT horizontally compressed.

Actually, it is. The ln graph is shaped in such a way that compressing it horizontally makes it look as if its been vertically shifted.

Last edited by JaneFairfax (2007-03-18 11:27:19)

Patrick · 2007-03-18 18:09:19

mikau wrote:

Likewise, with a function like ln(x^2) you'd think the graph would accelerate at a much faster rate than the graph of ln(x), since the argument is accelerating instead of moving at a constant rate. Instead, it just moves at a constant 2 times as fast. Creepy or what?
It seems everything related to e is surrounded by unique, amazing properties!

not very surprising is it?

This is just the case

Last edited by Patrick (2007-03-18 18:10:33)

mikau · 2007-03-19 05:58:02

JaneFairfax wrote:

mikau wrote:
but if f(x) happens to be ln(x), then ln(2x) is NOT horizontally compressed.
Actually, it is. The ln graph is shaped in such a way that compressing it horizontally makes it look as if its been vertically shifted.

It is and it isn't. If you consider the graph still spans from 0 to infinity on the x axis and it has the exact same shape in all areas. You can compress it, but it ends up just raising the graph. Its as if the graph is immune to compression!

It is not "suprising" that these things should occur, given the properties of logarithmic functions, ln(ax) = ln(x) + ln(a) and ln(x^n) = n ln(x), but when you consider how these changes work on a graph, its pretty amazing that the properties for functions still hold true, logarithmic rules being as strange as they are. It almost seems impossible! Crazy that an accelerating argument could only stretch the graph vertically by a constant factor.

Last edited by mikau (2007-03-19 06:00:26)

JaneFairfax · 2007-03-19 06:16:47

mikau wrote:

It is and it isn't. If you consider the graph still spans from 0 to infinity on the x axis and it has the exact same shape in all areas. You can compress it, but it ends up just raising the graph. Its as if the graph is immune to compression!

Its all down to how you visualize compression here.

For example, I can also claim that if f(x) = x, then f(2x) doesnt compress the original graph at all but merely rotates it anticlockwise by a small angle about the origin.

Last edited by JaneFairfax (2007-03-19 06:32:31)

mikau · 2007-03-19 08:13:10

right righto. I just think its uber cool that you could draw a curve, compress it, and be left with the same curve thats just a bit higher up! Thats just...wacky!

Last edited by mikau (2007-03-19 08:13:34)

Math Is Fun Forum

#1 2007-03-18 09:54:58

natural log function

#2 2007-03-18 11:16:04

Re: natural log function

#3 2007-03-18 11:25:58

Re: natural log function

#4 2007-03-18 18:09:19

Re: natural log function

#5 2007-03-19 05:58:02

Re: natural log function

#6 2007-03-19 06:16:47

Re: natural log function

#7 2007-03-19 08:13:10

Re: natural log function

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