Integrating a Function 1/x

[misogynisticfeminist]

Yeah, this questions may be a little elementary for some, but I don't seem to have any sources which would be able to tell me how do i integrate a function 1/x. Any help would be great.

: )

 

[arildno]

I assume that you know that an antiderivative is ln(x), but you don't know how to get there?

 

[Physics_wiz]

Go to www.google.com Type in "int(1/x)" and click "I'm Feeling lucky"

QED

 

[misogynisticfeminist]

I assume that you know that an antiderivative is ln(x), but you don't know how to get there?

whoa crap, ln (x) didn't cross my mind. Thanks alot...

 

[Physics_wiz]

ln(x) will never "cross your mind." You have to either memorize it or know how to derive it.

 

[da_willem]

$$x=e^y$$
$$\frac{dx}{dy}=e^y=x$$
$$(1/x)dx=dy$$
$$\int (1/x) dx=\int dy = y+constant = lnx+constant$$

 

[StatusX]

$$\int_1^x t^{-1 + \epsilon} dt$$

For $$\epsilon$$ greater than 0. Obviously this would be:

$$\frac{x^{\epsilon} - 1}{\epsilon}$$

And you can see by applying l'hopitals that this goes to ln(x) as $$\epsilon$$ goes to 0.

Now if you want to get more technical, and not assume anything, you can see that the derivative of a^x is:

$$\lim_{h \rightarrow 0} \frac{a^{x+h} - a^x}{h} = a^x \lim_{h \rightarrow 0} \frac{a^h - 1}{h}$$

Now if the limit exists, which you can see it clearly does by looking at the graph, this means that the derivative is some function of a times a^x. You can easily check that for a less than one, this function of a is negative, its equal to 0 at one, and its greater than 0 for a greater than one, and continues getting bigger and bigger. For some a, it will be exactly equal to one, and we'll call this e. So the derivative of e^x is just e^x. The inverse of exponentiation is logarithm, and ln(x) is defined as the log base e. Then the derivative of a^x is the derivative of e^(ln(a)*x) which you can see from the chain rule is ln(a)*e^(ln(a)*x), or just ln(a)*a^x. So the function of a was just ln(a). This can be used in the limit above:

$$\lim_{\epsilon \rightarrow 0} \frac{x^{\epsilon} - 1}{\epsilon} = \lim_{\epsilon \rightarrow 0} \frac{ln(x) \cdot x^{\epsilon}}{1} = ln(x)$$

Actually, you don't even have to use l'hopitals at this point, because if you look carefully you'll see this limit and the funtion of a are indentical, just with a replaced by x and h replaced by $$\epsilon$$.

 

[StatusX]

Actually, you can take this even farther and use it to find the value of e:

$$\ln(x) = \lim_{\epsilon \rightarrow 0} \frac{x^{\epsilon} - 1}{\epsilon}$$

Now noting that e^x is the inverse of ln(x):

$$y = e^x$$

$$x = \ln(y) = \lim_{\epsilon \rightarrow 0} \frac{y^{\epsilon} - 1}{\epsilon}$$

rearranging (and noting that these expressions are only valid in the limit as $$\epsilon$$ goes to 0):

$$\epsilon \cdot x = y^\epsilon - 1$$

$$y^\epsilon = 1 + x \cdot \epsilon$$

$$y = (1 + \epsilon \cdot x)^{1/\epsilon}$$

which leaves:

$$e^x =\lim_{\epsilon \rightarrow 0} (1 + \epsilon \cdot x)^{1/\epsilon}$$

or, plugging in 1 for x, and replacing $$\epsilon$$ with 1/n:

$$e = \lim_{n \rightarrow \infty} (1 + 1/n)^n$$

Which is the standard way of defining e. I'm sorry if this is much more than you wanted. I never really thought about this stuff, just accepting it at face value, and your question prompted me to try to derive it myself. I was suprised how nicely it all works out.

 

[Chrono]

whoa crap, ln (x) didn't cross my mind. Thanks alot...

Don't forget that in calculus they use log[x] as the natural log, instead of ln[x].

 

[dav2008]

Don't forget that in calculus they use log[x] as the natural log, instead of ln[x].

Can you explain that...?

 

[misogynisticfeminist]

Hey, thanks for showing how the integral for 1/y was obtained, that was helpful...

: )

 

[Chrono]

Can you explain that...?

Pretty much every calculus book uses log[x] instead of ln[x] to represent the natural log with base e. Remeber that in high school they told you that if you had log[x] without a base it was understood to be 10? Well, that's not ture in calculus. Log[x] has a base e. Is that better?

 

[dav2008]

Pretty much every calculus book uses log[x] instead of ln[x] to represent the natural log with base e. Remeber that in high school they told you that if you had log[x] without a base it was understood to be 10? Well, that's not ture in calculus. Log[x] has a base e. Is that better?

I have never seen log(x) used as the natural log

 

[Chrono]

I have never seen log(x) used as the natural log

I see it used all the time. Even Mathematica uses it like that.

 

[DeadWolfe]

Yes, but only mathematicians.

I don't go near engineers, because they smell of booze and modernism, but I'm told that they use log to mean log to the base 10.

 

[Galileo]

It's just a notational convention, not a 'new' definition for the notation log(x) because it's calculus. In my book on statistical mechanics they also use log for ln.
The author writes:

Note: All logarithms in this book are natural logarithms - The base 10 logarithm is as much of a historical curiosity as a slide rule.)

My (university) calculus book uses ln, for log base e. Simply to separate it from the other bases.

 

[da_willem]

Yes, but only mathematicians.

I don't go near engineers, because they smell of booze and modernism, but I'm told that they use log to mean log to the base 10.

My statistical mechanics book (Kittel) also uses log to mean ln. It has something to do with clarity. ln can be confused easily as a product of l and n or something...

 

[BobG]

Note: All logarithms in this book are natural logarithms - The base 10 logarithm is as much of a historical curiosity as a slide rule.)

Base 10 log is used to measure gain in power amplification, used to measure sound levels, used to measure earthquakes, used to measure power of radio signals and gain of antennas among other things.

Even more importantly,
What do they mean slide rules are only a historical curiosity! I still use mine!

:grumpy:

And what's all this talk of smelling like booze Geez, I've got to find a stronger aftershave

 

[matt grime]

But as log to any base only differs by a multiplicative constant, it is merely a change of unit away from being log in any other base.

log_{10} is useful since it is, give or take a small number, the number of digits in the original number. but it's still only factor of 2 and a bit away from log proper...

as we were told:

log will always mean log base e from now on. After all, why on Earth would you want to use anything else?

 

[Physics is Phun]

why would log mean log_e ?? that's what we have ln for ! Everything I've learned says that ln is log_e and log is log_10.
As for the integration part you just have to remember that any equation with 1/x or x^-1 will have a ln in it's answer.
I alway remember by doing this problem.
what is the antiderivative of 1/chair ?
I'll let you figure it out.

 

[Galileo]

Base 10 log is used to measure gain in power amplification, used to measure sound levels, used to measure earthquakes, used to measure power of radio signals and gain of antennas among other things.

Which is actually curious. They should've used the natural logarithms.
Who was this dopey guy who invented this dB scale? I think it was Alexander Graham Bell, who invented the telephone.
Well, that means he was an engineer so that figures :rofl:

...I was just kidding BobG, no need to get your underwear into a wrinkle.

 

[devious_]

I've gotten used to using lg as log_{10} and ln as log_{e}.

 

[Chrono]

I just think it's safe to always assume log[x] as being the natural log, unless otherwise specified.

 

[vsage]

I think engineers are right in this case. Ln is the notation for log base e in other countries (and not log), so I am told by my spanish (from spain) linear alg. prof. They don't call it the natural log though, they call it the Naperian log. That's curious because Napier lived well before Euler ever conceived of e.

 

[Alkatran]

In my opinion log() should be base ten, like our entire number system.

It shouldn't be base e because ln() already has that value. What's the point of having two notations for the same thing when one of them used to do something different? The peopel trying to change it are just setting everyone up for troubles.

 

[matt grime]

However, Alkatran, log to mean base e is *already* an established convention in mathematics (whatever it is elsewhere is not important to us here, really) in many areas. If I have to choose between the convention adopted by, amongst others the University of Cambridge, and the opinion of one person on a web forum who isn't a mathematician I know which I wil pick. Much the same as I will always refer to it as the complex plane and not the argand plane (one is a high school affectation that you ought to grow out of).

 

[BobG]

Well, even slide rules devote more space to natural log scales than log_10 scales. They only have 1 log_10 scales and your 50's era power rules have up to 8 log_log scales (the common slide rule term for natural log scales).

But log_10 is so easy to use in your head to get a rough approximation.

If you've got 20 Watts in with a 17dB gain, you know you've really got a (20dB-3dB) gain. The 20 dB gain means you need to add 2 zero's to your input (2000 Watts) and the -3dB gain means you need to divide by 2. Your final output is about 1000 Watts.

Napier invented the log_10 scale. Seems kind of strange the natural log is sometimes called the Napierian log. (Actually, Napier's work led Euler to come up with e and some have considered calling e Napier's constant since Euler already has a constant named after him).

And as to engineers -

But it is best not to be intimate with gentlemen of this profession and to take the calculations at second hand, as you do logarithms, for to work them yourself, depend upon it, will cost you something considerable.-William Thackeray

And this Thackery guy says this to every female I've ever encountered.

 

[Chrono]

However, Alkatran, log to mean base e is *already* an established convention in mathematics (whatever it is elsewhere is not important to us here, really) in many areas. If I have to choose between the convention adopted by, amongst others the University of Cambridge, and the opinion of one person on a web forum who isn't a mathematician I know which I wil pick. Much the same as I will always refer to it as the complex plane and not the argand plane (one is a high school affectation that you ought to grow out of).

I take it you prefer the natural log other than log to the base 10.

And were you talking about me as the person on the web forum who isn't a mathematician? Because I am working on that...somewhat. :tongue:

 

[matt grime]

No, I was referring back to Alkatran directly who opined that we shouldn't go changing to something, something that is alread established anyway in a large and important section of the *MATHS* community. Engineers can use base 10, whatever they feel like. Notational convention isn't universal. Deal with it, is the simple advice. I can think of three separate notations for dual in one book alone: D * and ^. All because of different historical precedences. It isn't confusing really, and should be rather obvious from the context. log 10 may be useful for numerical caclulations in some cases but isn't analytically any good.

 

[Fritz]

This calculus notation is so damn confusing.

 

[mathwonk]

to me, integral means limit of riemann sums, so integral of 1/x means just that. i.e. it means area under the graPH OF Y = 1/X.

now it is a theorem that this area function has a derivative which equals 1/x, and it is also a theorem that this area functioin behaves like a logarith, hence must be one, but all this is a long story.


By the way I love the following proof:

"Now if the limit exists, which you can see it clearly does by looking at the graph,"


I had always though it difficult to prove this limit exists! Another equivalent argument would be perhaps "which is clear from sticking your finger into the wind,.."

 

[StatusX]

it wasn't a rigorous proof. to anyone familiar with the graph, you know that it is differentiable. to prove this would be tricky, but all i meant to do was show why e is involved in this integral at all, and i did that.

 

[BobG]

By the way I love the following proof:

"Now if the limit exists, which you can see it clearly does by looking at the graph,"


I had always though it difficult to prove this limit exists! Another equivalent argument would be perhaps "which is clear from sticking your finger into the wind,.."

And your point is? :rofl:

It's in the fine tradition of Pierre Laplace, patron saint of math teachers. As Laplace's translator, Nathaniel Bowditch once said, "I never came across one of Laplace's 'Thus it plainly appears' without feeling sure that I have hours of hard work before me to fill up the chasm and find and show how it plainly appears."

Personally, I just go with George Castanza's "Yada yada yada ..."

 

[SomeRandomGuy]

I'm a mathematics major and we don't use log(x) to mean natural log. Maybe some professors have certain biases over others.

 

[shmoe]

What sort of math courses have you taken so far though? I don't think I've seen log to mean base 10 outside of high school or some texts used in the initial calculus stream (and calculators too I suppose). log is pretty much universally accepted to mean base e, at least when mathematicians are talking to one another (and not first year calculus students).

I haven't seen anyone mention base 2 yet. In some cryptography papers I've read published in computer science journals they used log to mean base 2. It was a convenient choice, but a standard that confused me when I first encountered it.

About the "ln" notation, most of you probably used it in your first calculus course at least. I was taught to pronounce "ln" as "lawn", as were most other students I've run across. I've recently been told this was a Canadian thing and that Americans don't do this. So I've been wondering how other countries teach you to pronounce "ln".