# Math Help - logarithmic equation

1. ## 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

2. Originally Posted by dfodor
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.

3. 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?

4. 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 !

5. 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

6. 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

7. 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 ...

8. After multiple steps of iteration and fitting you can define two parameters. But now I have no clue how to make it.

9. 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.