May 4th, 2008, 01:21 PM  #1 
Newbie Joined: May 2008 Posts: 9 Thanks: 0  equation
Anyone know how to solve a^x = bx? I know you have to put it in Ae^A = k where A = ln(a)x and k = ln(a) * b^1. Then you take Ae^A = k and say A = w(k). This works for one solution, but what about the other solution? Take 4^x = 8x. With the above method you get x = 0.15 which is right. But 2 is also right and how do you find it? I hope someone knows something about this because I really wanna know. Please forgive me if I posted this the wrong place. Thanks 
May 5th, 2008, 07:57 AM  #2 
Senior Member Joined: May 2007 Posts: 402 Thanks: 0 
Hm... First or all, please put ... on the end of your solution "x=0.15" because it's not correct otherwise.. And regarding your other question, you're trying to find the inverse or f(x) = 4^x  8x that's not bijective. So, without realizing it, you've limited yourself only to solutions x < 1.26438... That's why you're not able to get x = 2. Hope you get it!

May 5th, 2008, 08:37 AM  #3 
Newbie Joined: May 2008 Posts: 9 Thanks: 0 
Yes, sorry about that. 0.15.... of course! But what did you mean? I can't find 2? If you know a formula, please explain, I'm loosing my mind here 
May 5th, 2008, 10:30 AM  #4 
Senior Member Joined: May 2007 Posts: 402 Thanks: 0 
No, here's the deal.. When you've introduced the Lambert W function into your equation, you lost your x = 2 solution. It's quite similar when solving x^2=4. If you were to introduce the function f(x)=sqrt(x) then you'd get x = 2, but lack the solution x = 2. It's the same case here, but more complicated. Yet another example is solving sin x = 0. Using the f(x) = arcsin(x) function you get x = 0, but don't get an infinitive number of solutions that are gained by analyzing the sin function with more care. So, to answer your question  to get the second solution you'd need to use yet another LambertW like function.. It's just the trick of finding one! 
May 5th, 2008, 06:07 PM  #5 
Newbie Joined: May 2008 Posts: 9 Thanks: 0 
Please tell me if you know of one. I've really tried everything and I'm running out of ideas 
May 7th, 2008, 02:41 PM  #6 
Global Moderator Joined: Dec 2006 Posts: 20,627 Thanks: 2077 
The Lambert Wfunction is sometimes called the Product Log function, and is available in Mathematica as ProductLog[k, z], where k is 0 if you want the principal value (also available as ProductLog[z] or LambertW[z]), or 1 if you want the value which gives the second real solution for the above problem.

May 7th, 2008, 04:39 PM  #7 
Newbie Joined: May 2008 Posts: 9 Thanks: 0 
This is great! I feel I understand little by little. But could you please show me how to find 2 in 4^x = 8x for me? I think I could understand if you show me. I don't know much about functions, but I'm thinking maybe the other solution is the inverse of the first or something like that. Also I don't quite understand ProductLog[k, z]. What does the , do? Does it mean it's ProductLog from k to z and how do I use that in my equation? 
May 8th, 2008, 12:15 AM  #8 
Global Moderator Joined: Dec 2006 Posts: 20,627 Thanks: 2077 
As you know, 4^x = 8x can be rearranged as Ae^A = ln(4)/8, where A = ln(4)x. Hence x = w(ln(4)/8)/ln(4), where w is a suitable function. For real values of A, Ae^A is decreasing for A ≤ 1 and increasing for A ≥ 1. For the second range, the inverse is called the principal value of the Lambert Wfunction and denoted by W; in Mathematica software, W(z) is implemented as LambertW[z] or ProductLog[z], but ProductLog can also accept a second parameter (that is an integer written before z and separated from it by a comma). ProductLog[0, z] is equivalent to ProductLog[z], both giving W(z). For the first range, the inverse function, w, that you need is implemented in Mathematica as ProductLog[1, z] and ProductLog[1, ln(4)/8] has the value 2ln(4). Hence x = w(ln(4)/8)/ln(4) simplifies to x = 2. If you don't have the Mathematica software, you can solve Ae^A = z by an iterative successive approximation method of your choice, such as NewtonRaphson. 
May 8th, 2008, 05:41 AM  #9 
Newbie Joined: May 2008 Posts: 9 Thanks: 0 
Ok, that was simple But why does W[1, ln(4)/8] become 2ln(4)? If it said W[1, ln(16)/6], then that would become 8ln(3) or what? Thanks for the great info, and I use the wolfram site to solve Wfunction so that's no problem. 
May 8th, 2008, 06:18 AM  #10  
Global Moderator Joined: Nov 2006 From: UTC 5 Posts: 16,046 Thanks: 938 Math Focus: Number theory, computational mathematics, combinatorics, FOM, symbolic logic, TCS, algorithms  Quote:
 

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