Natural Logarithmic Function Definition

Jimmy King

I don't understand the definition of the natural log function. My calc book says ln x = integral from 1 to x of (1/t) dt. What is this "t" and where did it come from? This is not explained anywhere. I also haven't seen it explained anywhere else I've looked this up. It seems everything expects that I should already be familiar with whatever t is intended to mean here.

minirhyder

dt means you're integrating with respect to t, that's where the t comes from. It's just another variable so you won't confuse it with x.
The x comes in later when you plug it in after you've integrated 1/t.

Jimmy King

I understand integration, I just don't know where t comes from or what it would be equal to. I don't understand its relationship to x.

ln5 = integration from 1 to 5 of (1/t) dt What is the value of t here?

minirhyder

You would integrate (1/t)dt and would get log(|t|). You would then plug in the limits of integration (1 to x) and get log(|x|) - log(|1|).

The t is just an arbitrary variable used to express the integral, and the x is a limit of integration.

It would be the same exact thing to say ln(a) = integral from 1 to a of (1/x)dx.

BlazeFire

Here is another way to think of it: The value of ln(x) is the area under the curve described by the function f(t) = (1/t) between 1 and x.

It isn't so much about the 't' specifically as it is the function f(t).

Let's say we have a made-up function called the unnatural logarithm, written as lu(x). The value of lu(x) is the area under the curve g(t) = (t^2) between 0 and x.

Again, 't' is just being used to describe the function 'g'.

Jimmy King

Hm. I kind of see. I get what you're saying, I'm just not quite sure how it's relevant at this point in my book, as in I don't yet understand how I'd use that. I guess I'll keep plugging along and come back to it. I have this trouble with math a lot. I learn much better from application and seeing real examples and then working backwards, but math is so rarely taught that way. Getting these abstract formulas and no actual anything to plug in usually just confuses the hell out of me while I try to figure out how to apply them to the next 3 pages of stuff that may or may not be directly related.

BlazeFire

Are you okay with the idea that ln(x) equals the area under a specific function between two points?

minirhyder

To illustrate @BlazeFire 's point

Jimmy King

Ah hah. I think it clicked. They're giving me a function... f(t) = 1/t and then lnX is just the integral of f(t) from 1 to X?

BlazeFire

Yup! That is the key.

Spend a few minutes looking at that again to get it settled in. It isn't uncommon that one function (in this case, ln(x) ) is defined as the result of an integral where x is one of the limits. The first example that comes to mind is the error-function (a probablity/statistics thing).

Jimmy King

Jesus. Why the hell did the book take 2 paragraphs and 2 highlighted formulas, yet nowhere is f(t)=1/t mentioned. Now that I realize that it's obvious that it's implied because that's what an integral is, but since there has never been a function on both sides of the equation yet in this book, my brain didn't process that at all. If they would have just spelled it out a bit more it would have clicked much sooner.

Thanks for the help guys. I'm sure I'll be back again with more stupid questions.

BlazeFire

Not many stupid questions when it comes to calculus.