Naturual Logarithms

Bob · 2010-08-23 03:46:21

Hi Dave and Bobbym

As long as you fix the domain of one to be the range of the other, a function and it's inverse will be mirror images of each other in the line y=x. (see diagram).

Bob

Last edited by Bob (2010-08-23 03:51:19)

Neetu · 2010-08-24 00:26:04

hi all

can anybody help me in solving the following problem

ln(x)+x=constant

solve it to find x. please help me out. Thanks.

Neetu · 2010-08-24 00:28:27

sorry, above equation is ln(x)-x=constant

bobbym · 2010-08-24 00:30:53

Hi Neetu;

Welcome to the forum.

I don't believe it can solved in terms of elementary functions. You would to have use a product log or Lambert function I believe.

Neetu · 2010-08-24 00:35:31

thanks

but if you can solve this equation by any means, that will be very helpful for me, as m working on project and to proceed further i need to solve this equation.

bobbym · 2010-08-24 00:37:35

Hi;

You cannot get an analytical solution in terms of elementary functions, To get an answer for some problem you will have to define a, such as a = 2 and solve numerically. That can be done. You can represent the above function as a Taylor series also.

You will not solve in some form like this x = (2a - 3) / 5. From the conditions of your problem determine what a is or an interval for a. This can sometimes be done if you have formulated the problem well. Then the solutions can be gotten in many ways.

Neetu · 2010-08-24 00:53:02

a=2, that means you are asking to define constant?

constant can be nynumber in that equation but shud b >=0

so equation can be wriiten as ln(x)-x=2, it doesnt matter

actully the main problem is ln(x1*x2/x3) - x1*x2/x3 = constant

and i need to find x1 in terms of constant, x2 and x3

bobbym · 2010-08-24 00:57:39

x1 can be solved in terms of x2 and x3 but smack dab in the middle of it will be the lambert function.

Where W is the Lambert function. Same problem as above.

Neetu · 2010-08-24 01:03:22

ok thanks.

you are sayin x1=x3W(e^(-a))/x2 is same as ln(x1*x2/x3)-x1*x2/x3=a?

Neetu · 2010-08-24 01:17:17

can you also tell me W(e^(-a)= ?

like W(e)=1, W(-e^(-1))=-1

can you help me on solving W(e^(-a))?

Bob · 2010-08-24 01:19:52

Hi Neetu,

If ln(x) - x = 2 (I'll use 2 for the moment) then ln(x) = x + 2 .

Because the function 'ln' is so very different from x + 2 there isn't an easy mathematical way to solve this equation.

So I thought, would it be acceptable to graph y = ln(x) and also y = x + 2 and see where they cross. The diagram below shows what happened.

The graphs don't cross and never will because ln(x) is only positive after x = 1, by which time y = x + 2 is well above the axis and the log graph gets bigger at a slower and slower rate. This means it will never catch up to cross y = x + 2.

If you make the constant bigger the same thing happens.

We might be able to help if you go back a few steps in your project and explain how you got to this equation. Maybe there's a way to re-define the problem in terms that can lead to a solution.

Bob

Akpolome · 2010-08-24 01:50:13

ò y ' dx = ò (1/2) sin(2x) dx

Neetu · 2010-08-24 04:23:40

Hi bob, thanks for tryin to help me

main equation is : L (Pnk, αnk) = ∑_k= 1 toK▒ ∑_n=1 to N▒ αnkBlg(1+ Pnkgnk/N0B αnk)
+ lk (∑_(n=1)^N▒ Pnk - Pk) + βn (∑_(k=1)^K▒ αnk 1)

where lg defines log to the base 2

after takin dertivate of above equation w.r.t ank and eqauting equation to 0, we get

ln (Pnkgnk / ank) - Pnkgnk / ank = βn1

where βn1 is constant relative to βn

i want to find pnk in terms of gnk,ank and βn1

please help me in solving this equation.

Bob · 2010-08-24 04:55:24

Thinking ....................

Bob

Neetu · 2010-08-24 06:31:20

the above problem can be relaxed.

i need to find pnk, main aim is to have pnk at left hand side, it can be in any terms. it can be in terms of βn, B, ank, gnk, No*B....what ever.

Bob · 2010-08-24 08:07:26

Hi Neetu,

I'm having lots of trouble getting anywhere with this.

Problem (i) : Your 'main equation' has an unrecognisable (to me ) symbol in it. It occurs 4 times; always after the K or N.

Problem (ii) : I think some of your variables are shown with 3 symbols eg Pnk. This confuses me as I cannot tell which are single variables and which are several multiplied together.

eg.

Problem (iii) : If I ignore the above and go back to

I still have the problem of no real solutions. (See revised graphs below)

y = x + positive constant will always be above the x axis and have gradient = 1

y = ln(x) start below the x axis and only crosses at (1,0) with gradient = 1. Thereafter, the gradient reduces.

So these graphs will never cross in real numbers. So no real solutions to the equation exist.

Are complex solutions allowed in this project?

There is a way of showing mathematical symbols with 'proper' layout by using the language LaTex. I only started to use it today and already have managed to show integrals with limits, nested brackets and fractions. I have used it above to show the problem with subscripts. The instructions for it are in the top post in the 'Help Me' section. This might help you to show your main equation in the way you would usually write it.

You can look up the W function at http://en.wikipedia.org/wiki/Lambert_function

but I suspect it will just add to your problems.

For the moment that's all I can do. Sorry.

Bob

Last edited by Bob (2010-08-24 08:17:58)

bobbym · 2010-08-24 11:45:50

Hi Neetu;

can you help me on solving W(e^(-a))?

That can't be done. It is like asking me to solve for sin(a) or tan(a) or e^(a). Need to know what a is. Here is some general advice. You have an answer it is terms of the Lambert function. Who knows what that is returning, possibly a complex value. You can now get an answer for any a, as long as you know what a is. What interval? You need information for a. Once you say a = .35 then I can solve for x. But only numerically. Very few equations can be solved by algebraic manipulations. School makes you think that wow for all equations we can solve for x. This is true only for 1,2,3,4 degree polynomials. Equations with transcendental functions ( log, sin cos e^ etc...) are usually very difficult or impossible, especially for linear factors.

Here is another point you have 1 equation and 2 variables a and x. You are asking what is a or what is x. To solve for one of the variables I would need another equation. Hopefully you see your problem as posed is leading to dead ends. You must formulate it better.

Neetu · 2010-09-17 12:46:46

thank u guys... u tried ur best....and with ur help i achieved some where...... thank u so much... u r doing gr8 job...keep it up.

my prob was just to solve that equation, which is solved by ur given soln as follows:

x1=x3W(e^(-a))/x2

u meant above eqn is same as ln(x1*x2/x3)-x1*x2/x3=a

Am i right?

can u also tell me...what will be lambert function of a (costant). i mean will it be some constant as well?

can u plz also describe why we need to use lambert function as i need to put all this in report.

once again thank u so much!!

bobbym · 2010-09-17 18:11:51

Hi

Not exactly:

Is a rearrangement of the other equation.

Neetu · 2010-09-19 06:49:04

thanks. but i have used following equation in my project

x1=x3W(e^(-a))/x2

which u given earlier. i wanted to kno wht W(e^(-a)=?, if 'a' is positive. wil it b positive?

please help me out.

bobbym · 2010-09-19 07:15:01

Hi Neetu;

Whoa, hold on. That equation is not correct. Please use the equation in post #44. I have looked back on the work and see the same equation in post #33. I probably edited it a couple of times after I posted that, but I can't remember. Keep this in mind about answers on forums, they are a work in progress and my typing is so bad that I have to edit them 6 - 7 times. It takes me a long time to complete a post so please wait for me to finish a post. Especially before copying it anywhere. That will save us both a lot of heartache.

Anyway post #44 is the correct answer so you don't have to consider the other form.

Neetu · 2010-09-19 07:34:35

hi..

but cant i use eqn on forum #33? and can you explain y i cant use it in reference to forum #44

bobbym · 2010-09-19 07:38:27

Hi

This

and the one you want to use

Are not the same thing. Not by a longshot.

Neetu · 2010-09-19 09:38:24

yes, i want to use 2nd one but there is no -ve sign. please see equation in ur forum #33

bobbym · 2010-09-19 09:53:10

Hi;

Here is what is in post #33

It looks like the first one. You can not use the second because it is wrong.
The second one looks like a typo, but I can not remember.

Math Is Fun Forum

#26 2010-08-23 03:46:21

Re: Naturual Logarithms

#27 2010-08-24 00:26:04

Re: Naturual Logarithms

#28 2010-08-24 00:28:27

Re: Naturual Logarithms

#29 2010-08-24 00:30:53

Re: Naturual Logarithms

#30 2010-08-24 00:35:31

Re: Naturual Logarithms

#31 2010-08-24 00:37:35

Re: Naturual Logarithms

#32 2010-08-24 00:53:02

Re: Naturual Logarithms

#33 2010-08-24 00:57:39

Re: Naturual Logarithms

#34 2010-08-24 01:03:22

Re: Naturual Logarithms

#35 2010-08-24 01:17:17

Re: Naturual Logarithms

#36 2010-08-24 01:19:52

Re: Naturual Logarithms

#37 2010-08-24 01:50:13

Re: Naturual Logarithms

#38 2010-08-24 04:23:40

Re: Naturual Logarithms

#39 2010-08-24 04:55:24

Re: Naturual Logarithms

#40 2010-08-24 06:31:20

Re: Naturual Logarithms

#41 2010-08-24 08:07:26

Re: Naturual Logarithms

#42 2010-08-24 11:45:50

Re: Naturual Logarithms

#43 2010-09-17 12:46:46

Re: Naturual Logarithms

#44 2010-09-17 18:11:51

Re: Naturual Logarithms

#45 2010-09-19 06:49:04

Re: Naturual Logarithms

#46 2010-09-19 07:15:01

Re: Naturual Logarithms

#47 2010-09-19 07:34:35

Re: Naturual Logarithms

#48 2010-09-19 07:38:27

Re: Naturual Logarithms

#49 2010-09-19 09:38:24

Re: Naturual Logarithms

#50 2010-09-19 09:53:10

Re: Naturual Logarithms

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