Natural Definition and 1000 Threads

I want to verify this: 2ln(x)-ln(2x)=ln(x^2)-ln(2x)=ln\left(\frac{x^2}{2x}\right)=ln\left(\frac x 2\right) ln(2x)-ln(x)=\ln\left(\frac {2x}{x}\right)=ln(2) Thanks

So I have an exam tomorrow, and the teacher provided a review. f(x) = ln(x + y) I remember that d/dx ln[f(x)] = f'(x)/f(x) so would that not equal 2/(x + y) ? The answer she gave is 1/(x + y - 1) ... where that neg. one came from I have no idea. Come to think of it, there were no...

Hope this is in the right place... I'm trying to understand why the derivative of ln(x) is 1/x while the derivative of something like ln(4) is 0. My knee-jerk reaction is to view 4 as representative of x, thereby giving me F'(x) ln(4) = 1/4, not 0. That would be the case, except ln(4) is a...

"Quantum Optics," by Scully and Zubairy, attempts to explain the natural line width of an emitting atomic ensemble (in a resonant cavity with leakage) using random spontaneous emission events of the atoms (which in turn is due to interactions with the vacuum field). The basic idea and...

Given n numbers x1,x2...xn belong to N. x1+x2+x3...xn=m How many different combinations of x1,x2,x3...xn are there?Is there any formula useful here? Note:x1,x2,x3... need not be distinct and also can be 0. Thanks

Homework Statement \int^{1}_{0}\int^{e^x}_{e^-x}\frac{lny}{y}dydx The attempt at a solution So I am integrating ln(y)/y and I tried it by parts, first with u = ln(y), dv = 1/y, and therefore du = 1/y, and v = ln y but if I use that I get (ln(y))2-\int\frac{lny}{y} again. So I tried...

When deriving the natural level of output i am confused as to how the leap is made in the last two lines. P = P(1+μ)F(1-Y/L,z) => 1 = (1+μ)F(1-Y/L,z) => Yn = h(μ,L,z) I understand how the mark-up, Total Labour and variable factors are the factors of the natural level but how does...

Homework Statement The length L1, is not the wavelength of the fundamental frequency of the string. With the tension equal to F1, to which natural frequency does the wavelength equal to L1 correspond? Homework Equations I was reading online, and found that when a string vibrates at...

I figured I would just add this new problem over here, rather than starting a new thread. Im looking to solve integration leading to arctan or arcsin results. \int_{1}^{e}\frac{3dx}{x(1+\ln(x)^2}) Looking at this, it feels like this has an arctan in the result, but I would have to multiply...

The natural logarithm can be expressed as lim n(x^(1/n)−1) n->ifinity Can someone please derive this for me? I've tried using the limit definition of e^x and then applying f^-1(x) but i can only get ((n*x)-1)^1/n any help is greatly appreciated thanks!

There are 3 long lived (over 10^9 years) isotopes which may capture electrons and have actually been observed to undergo single beta transmutation. Interestingly, all these 3 ALSO can undergo beta decay AND positron emission. Potassium 40 has a branching ratio of about 1000 electron...

please help i have to write a MATLAB code for DETECTION OF THE LOCATION AND SIZE OF A CRACK IN STEPPED CANTILEVER BEAMS BASED ON MEASUREMENTS OF NATURAL FREQUENCIES the reference paper is DETECTION OF THE LOCATION AND SIZE OF A CRACK IN STEPPED CANTILEVER BEAMS BASED ON MEASUREMENTS OF...

I have learned that beta-lactamase enzymes have very ancient origin. And they are just tinkered in the recent anthropogenic activity. So some original form must be out there. What are their natural target?

Not to mention that I am one of those who thinks that sitting on a metal chair and table is no big of a deal as long as we can preserve the natural beauty of this planet and also I don't like egoistic people who think only about themselves now this being said this video is both a great idea , a...

I've always thought that evolution was this mysterious property of the universe and that natural selection was almost a separate theory. Now that I look more into it, i can almost always rationalize an answer as to why a property of life is the way it is by natural selection. For example, i...

I have been given a question by my tutor to try out for our next class Using the axioms for addition of numbers give a natural language proof that the additive inverse of a number is unique, that is prove: ∀x∀y∀z (x + y = 0) ^ (x + z = 0) → (y = z) I am new at writing proofs! My...

Homework Statement ∞ Ʃ (-1)^(k+1) / kln(k) k=2 Homework Equations integral test, p test, comparison test, limit comparison, ratio test, root test. The Attempt at a Solution In class so far we have not learned the alternating series test so i can't use that test. So far I have...

EDIT: Found the answer, seems I overlooked part of the solution in the learning materials ( answer extended into another page) the Solution does indeed equal what i thought it did. Homework Statement So this is the problem i have: (2(-1)^n -((n*pi)^2(-1)^n)-2)*(8/(n*pi)^3) where n...

I'm trying to write down an axiomatic development of most of mathematics, and I'm wondering whether it's logically permissible to use natural numbers as subscripts before they have been defined in terms of the Peano Axioms. For instance... the idea of function is used in the Peano axioms...

What is twice the base of the natural logarithm?

If σ(N) is the sum of all the divisors of N and τ(N) is the number of divisors of N then what is the sum of sum of all the divisors of first N natural numbers and the sum of the number of divisors of first N natural numbers? Is there any relation between σ(N) and τ(N) functions? Can I do that...

Prove that a^3+b^3 \ne to \ c^3 \ if \ a,b \ and \ c \in \ {N} This is not a challenge,I am asking for help... Any help is appreciated... Thanks...

Homework Statement I've been puzzling with this for 1 hour and can't figure it out. How do you calculate the unknown x from this equation? Homework Equations ln(1-x)-5x+7.5=0

Hello, I was playing around with DeMoivre's formula ei*pi = -1 and there is something I don't quite understand about taking the natural logarithm of a certain expression. I though that e2i*pi = 1 ln[e2i*pi] = ln (1), but this yields to an imposibility 2i*pi = 0. So...

Homework Statement 2^{Sin(PiX)} Homework Equations The Attempt at a Solution = e^{(Sin(\PiX})ln2 y'= e^{(Sin(\PiX})ln2 d/dx [Sin \PiX] ln 2 and I'm stuck... The product rule will automatically lead to to differentiating ln 2.

Solve the following equation for x 16-12ln(7x)=47 I know I need to isolate x . however I am unsure how to get ln(7x) separated. what i have so far ln(7x)=-31/12

if f(x)= 3x lnx, then f ' (x)=? i used f' ' (x)=3x(D lnx) + D (3x) (lnx) f ' '(x)=3x (1/x) + 3 (lnx) so... f ' '(x)=3+3lnx or 3(1+lnx). unfortunately this isn't one of the possible answers given. could one of you kind folks help me understand where i went wrong? thank you

i am studying real analysis from terence tao lecture notes for analysis I. http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/ from what i understand , property is just like any other statement. for example P(0.5) is P(0) with the 0s replaced with 0.5 . so the notes says (assumes ?)...

I thought this area was appropriate for this question from a previous exam paper which I just need checking. So: Working out: The half-life of 244Hb is 10 million years. The half-life of 244Hb is 5.2595×10^12 minutes. The mean life of 244Hb is 7.58783677×10^12 minutes. So if you...

not really a problem, but more curious if we differentiate ln(2x) we get 2/(2x) = 1/x by the chain rule, but if we integrate 1/x we get ln|x|? Could anyone explain why this is the case, thanks.

Hi,Im just beginner and I m trying to learn integrals.I m just in starting phase,but still in few tences,not details...How or why we get logarithm in gibbs free energy equation?Because of integration of this equation or due to probability and statistics laws? Thanks

Homework Statement For all non zero natural numbers n prove that: 1- 24\mid n(n+1)(n+2)(n+3)(n+4) and that : 2- 120\mid n(n+1)(n+2)(n+3)(n+4)(n+5) The Attempt at a Solution 1- For n=1 we get that 24 divides 120 so we assume that 24 divides n(n+1)(n+2)(n+3)(n+4) and we...

I have a large volume of (free) natural gas sitting at 5,000 psi. I'd like to pass the gas through an orifice, give it room to expand on the backside, and use that to chill glycol. I would like to drop the output pressure to 100 psi in the process. I wish to use the chilled glycol to cool...

Okay, this is a really simple question, so to anyone looking for some extraordinarily complex differential equation question turn away now, or be blinded by boredom. My query is rooted in a question I had about building a water clock... so seemingly relevant to Differentials, I know...

I'm reading back over a calculus book getting ready for an exam and I'm seeing a note that I don't understand. It says to make sure, when rewriting a ln function that the domain is the same, then it provides an example of when it's not the same, yet says nothing more. Is this rewritten form...

I'm slightly confused about natural units. Take mass as an example: I can measure something in kg's, but then decide to convert to MeV/c^2, for instance. To do that I would multiply the quantity in kg by 3*10^8 squared and divide by 10^6*1.6*1-^(-19) i.e. the SI values of the constants. If I...

For instance, say I have -ln(-∞) Does the negative sign on the natural log cancel with the negative sign on the infinity? Is this true? -ln(-∞) = ln(∞) Thank you -Drc

Homework Statement The initial amount of radioactive atoms on a sample of 24Na is 10^10. It's half-life corresponds to 15 hours. Give the amount of 24Na atoms that will disintegrate in 1 day.Homework Equations I started to solve it using the formula N=Initial Amount of Atoms /...

The question is which sets of natural numbers are closed under addition. I know that odd is not, and I know how to prove that sets of multiples are, but my professor said there is something more and that is has to do with greatest common divisor. He said to pick numbers like 3 and 5 or 5 and 8...

I recently struck a question that I have not been able to find an answer to. I feel like I'm missing something obvious, so I've come here for help. The derivative of a^{x} is a^{x}lna. The explanation that Stewart 5e gives is: \frac{d}{dx}a^{x} = \frac{d}{dx}e^{(lna)x} =...

The function f is continuous at some point c of its domain if the limit of f(x) as x approaches c through the domain of f exists and is equal to f(c). In mathematical notation, this is written as lim_{x\rightarrow c} f(x) = f(c) from the positive and negative sides . For ln(x) (the natural log...

Homework Statement Ok, basically I need to show that Ʃ 1/n (between 1 and n) (which is harmonic number) is θ (big theta) of ln(n), which means that is it bounded below and above by this function(upper and lower bound). But I don't quite understand how to prove it.Homework Equations I know...

Homework Statement Ok, so I have the problem, and I have the answer, but I don't know how they arrived at the answer. I'll show you the problem, and what I tried and maybe somebody can point out where I am going wrong, or some hints to put me on the right path. (2x-1)/(x+1) being integrated on...

Resonance frequency and Natural frequency?? Say there is an empty bottle of coke., and you know for sure that the bottle resonates most at 320hz. when the bottle resonates, the bottle will vibrate. the question is, 1.if the bottle also resoates at 318hz( bu not most), will the bottle...

I am just wondering what is the proof of the equation 2*pi*f = 1/√(LC)

Homework Statement Consider the spring-mass system shown. A dynamic load, P(t), is applied with a frequency of 300 rpm. The helical compression spring is made of a music wire. When installed, the spring force is to vary between 100 and 500 N while the spring height varies over a range of...

Hello! I was trying to look for a possible expansion of the ln function. The problem is, that there is no expansion that can be used in all points (like there is for e, sine, cosine, etc..) Why do you think that is? To clarify: Let's say i do the MacLaurin expansion of ln(x+1)...

The integral of 1/x is ln(x). Where does that come from? That always puzzled me. We can continue to take derivatives through x^0 and into the negative integers, and just use the plain old power rule to get the answers. We can do the same for the integral of x all the way from negative...

Homework Statement This is to help out a 40something calc student -- thank you all in advance for your help Homework Equations If f (y) = ln ln ln x, what is ∂y/∂x? The Attempt at a Solution I came up with 1/x, which I got by applying ∂y/∂x ln x = 1/x three times, is this...

Let T : R2 -> R2 and S : R2 -> R2 be linear transformations defined by: T(x; y) = (5x + y ; 2x + 2y) and S(x; y) = (3x + 2y ; x): (i). Find the image of the line 2x + 3y = 5 under T. (ii). Find the natural matrices of the linear transformations T o S and T^-1 Sorry, I haven't done...