Integral of 1/x: Proving Invalidity of Method

[alice22]

Actually, convenience is an acceptable answer. Much of what mathematicians do is based off of convenience. Just think about the table of integrals in any calculus book. A mathematician could solve any of those functions by himself every time he needed them, but why would he when he could just solve the general case and use it?

It's the same general idea of convenience, picking the number that makes things easier.

Also, just out of curiousity, is it possible to define the sine function as an integral of the cosine function? Just a little side question.

You can define cos as tan if it makes the problem easier, after all convenience is what it is all about, who cares if it makes any sense?

 

[losiu99]

We can define things the way we want. If we define x as the number whose square is three, "x" will mean all the objects with this property. If such notation proves useful, it may become standard. Just like "pi" is defined to be ratio of circumference to diameter. Why doesn't "pi" mean 3? Because we defined it not to.

Next, I don't imply you have problem. You mentioned "answer" in you post, so I just asked "answer to what question?". You don't want to be just told the answer, that's good. But in order to help you understand the answer, I need to know the question. I'm obviously missing your point, so it would be helpful if you clarified it.

I'm not sure what you mean by "stop at one". If you mean that it's defined for positive reals less than one, you're obviously right. It has nothing to do with lower limit of the integral, though. We don't have to start integrating "from the very left".

Edit: Sure we could define cos as tan. There was no point, however. As long as convenience doesn't collide with validity, there's nothing wrong about it. We want definitions to be as useful and comfortable as it's possible, don't we? The whole point of defining is about convenience. We call some specific subsets of $$X\times Y$$ a function, so that we don't have to describe this desired properties every time.

 

[Mute]

You can define cos as tan if it makes the problem easier, after all convenience is what it is all about, who cares if it makes any sense?

As I said in the post at the top of the page, choosing the lower limit to be 1 makes the integral definition coincide with all of the other definitions of the logarithm. It would otherwise differ by a constant. Even if the integral definition came first, once the other definitions became known (and were found to be more useful), someone would have modified the definition to have the constant 1 so as to make the integral definition match up with the other ones.

If you don't find that to be a satisfactory answer then I don't know what answer you could possibly be looking for.

 

[Mark44]

No I have problems with that.
For starters show me the mathematics equation for "someone wanted something"?
I do not recall covering the that in my math course, maybe I missed the maths of "someone wanted".

There is no mathematics equation for "someone wanted something" but there are many equations that arose because someone wanted to find out something.

An equation such as x - 2 = 0 is very simple to solve, but a similar equation, x + 2 = 0 is impossible to solve if you don't understand the concept of negative numbers, which came along only about a thousand years ago.

Also, if you know that 3*3 = 9, you can guess at a solution to x² = 9, but if you don't have the concept of irrational numbers, you'll have a tough time solving the equation x² = 3.
[/quote]

example x^2 = 3 (because someone wanted it to be).

The inverse of the exponential function does not stop at one so that again is a wholely invalid reason.

What does that have to do with anything? Numerous people in this very long thread have given you perfectly valid reasons for the integral definition of the natural log function to be as it is. If you have a question about what they said, ask it, but do not wave your hand and say that these explanations are "wholely (sic) invalid" merely because you don't understand them.