Solve complex exponential equation

[Smed]

I'm having some trouble solving for t in the following exponential equation.

$$ B = A_1 e^{-\lambda_1 t} + A_2 e^{-\lambda_2 t} $$

I can't divide out the leading coefficients A1 and A2 because they differ. I'm not really sure how to immediately take the natural logarithm of both sides since the rhs would just become,

$$ \ln({A_1 e^{-\lambda_1 t} + A_2 e^{-\lambda_2 t}}) $$

Any help is appreciated.

 

[chiro]

Hey Smed and welcome to the forums.

How much mathematics have you taken?

Usually for general problems, we don't often have an analytic solution, or the analytic solution is either unknown or too complicated.

In the above case we use what is known as root-finding algorithms like Newtons method to solve the root of an equation which would give you a value.

If you can't find an analytic version easily or at all, try and use a numerical routine to calculate a good enough approximation of t which should suit your purposes. The value of t won't give exactly 0, but it will be close enough depending on what you calculate and how accurate you want it to be.

 

[Smed]

Chiro,

I've taken enough mathematics that I probably should be familiar with root-finding algorithms, but I wasn't. I think part of my problem is that I'm not sure what makes the equation not have an analytical solution. It seemed that way after trying to solve it for a while, but I figured I was just missing some simple technique.

After reading up on the Newton method, I was able to solve the problem iteratively.

Thanks for your help.