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Math Help - logarithmic equation

  1. #1
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    logarithmic equation

    Hi,

    I cannot solve the following equation and would like to ask for some help. The solution is important of course but I would like to see the steps or would like to get some hint how to solve it:

    ln(a0/a)+c(a0-a)=bct

    known are a0 and t, b and c are parameters and I need to solve it for a.

    Could anyone help please? I know it is a newbie question but I got stuck.

    Thanks

    dfodor
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  2. #2
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    skeeter's Avatar
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    Quote Originally Posted by dfodor View Post
    Hi,

    I cannot solve the following equation and would like to ask for some help. The solution is important of course but I would like to see the steps or would like to get some hint how to solve it:

    ln(a0/a)+c(a0-a)=bct

    known are a0 and t, b and c are parameters and I need to solve it for a.

    Could anyone help please? I know it is a newbie question but I got stuck.

    Thanks

    dfodor
    \ln\left(\frac{a_0}{a}\right) + c(a_0 - a) = bct

    the value of a in this equation cannot be solved using elementary algebraic methods.
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  3. #3
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    thanks, thats what I thought but as I have no math skills I was not sure. The a-s are measured values. Is there any way to find the parameters with some numerical method?
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  4. #4
    Super Member Bacterius's Avatar
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    Is there any way to find the parameters with some numerical method?
    What are you seeking to do ? Solve for a or for the parameters ? Anyway, as Skeeter said, there is no elementary (in other words, easy) way to solve this equation for a. It might even be impossible algebraically, eh !
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  5. #5
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    If you know numerical values for everything except a, a_0, b, c, and t, then, yes, you could use a numerical method like Newton-Raphson.

    I might also point out that by taking the exponential of both sides of ln(\frac{a_0}{a})+c(a_0-a)=bct we can change the equation into \frac{a_0}{a}e^{ca_0}e^{-ca}= e^{bt}. Multiplying both sides by ae^{ca}e^{-bt}, we get ae^{ca}= a_0e^{ca_0- bt}.

    Now, let x= ca so that a= x/c and we have xe^x=  ca_0e^{ca_0-bt}. Now we can solve that with the "Lambert W function", defined as the inverse function to f(x)= xe^x: x= W(ca_0e^{ca_0-bt}) and, since a= x/c, a= \frac{1}{c}W(ca_0e^{ca_0- bt}).

    A method of numerical approximation to the Lambert W function is given here:
    http://en.wikipedia.org/wiki/Lambert...#Approximation
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  6. #6
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    Thank you guys for your replies. So it is a model to describe reaction kinetics and b and c parameters are needed to characterize the reaction. a0 is the initial concentration, a is current concentration measured at t minute. b and c are unknown. I would need a numerical method to determine these two parameters, all other variables are known. First I wanted to solve it for "a" to have theoretical values but more important is the estimation of the parameters based on experimental data.

    I will go through on your posts and try to solve the problem, but I would be thankful if you could help.

    Best regards

    dfodor
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  7. #7
    Super Member Bacterius's Avatar
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    b and c are unknown. I would need a numerical method to determine these two parameters
    So, I understand you want to find ( a or not), b and c with only one equation ? This is definitely not possible. You need at least another equation involving one of the unknowns, otherwise there are infinitely many solutions. Maybe this "experimental data" could be useful ?
    Or perhaps I am just stupid and don't understand what the question is all about ...
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  8. #8
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    After multiple steps of iteration and fitting you can define two parameters. But now I have no clue how to make it.
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  9. #9
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    Does it make sense to invert the function so that t is my dependent variable? I get believable parameters and good fit. As this model is quite common and it is nowhere to read that it was approximated with Lambert W function or so.
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