# Thread: Equation involving sum of exponentials

1. ## Equation involving sum of exponentials

-12.5exp(-1500t) + 16exp(-2400t) + exp(-600t) = 0

solve for t. I don't know how to take the natural log of a sum of exponentials. thanks for your help!

2. There may be an easier way, but here goes:

Starting with

$\displaystyle -12.5 e^{-1500t} +16 e^{-2400t} + e^{-600t} = 0$

Divide through by $\displaystyle e^{-600t}$, and rearrange:

$\displaystyle 16 e^{-1800t} -12.5 e^{-900t} +1 =0$

This factors out to the form:
$\displaystyle (Ae^{-900t} -1)(Be^{-900t}-1) = 0$ (eqn 1)

Where $\displaystyle AB = 16$ and $\displaystyle A+B = 12.5$. You have two equations in two unknowns, which gets you to:

$\displaystyle A^2 - 12.5A +16 = 0.$

Use the quadratic formula to solve for A:

$\displaystyle A = \frac {12.5 \pm \sqrt{12.5^2 -64}} 2 =$ 1.448 or 11.052, B = 12.5-A = 11.052 or 1.448

Put these values back into the equation (1):

$\displaystyle (1.448e^{-900t} - 1) (11.052e^{-900t} -1) = 0.$

So:
$\displaystyle e^{-900t} = \frac 1 {1.448}$
$\displaystyle t = \frac 1 {900} ln(1.448) = 0.000411$

or:
$\displaystyle e^{-900t} = \frac 1 {11.052}$
$\displaystyle t = \frac 1 {900} ln(11.052) = 0.00267$

3. awesome

4. Hello, listonroute!

Another approach . . .

Solve for $\displaystyle t\!:\;\;-12.5e^{-1500t} + 16e^{-2400t} + e^{-600t}\; =\; 0$

I don't know how to take the natural log of a sum of exponentials.
Nobody does!

We have: .$\displaystyle e^{-600t} - \frac{25}{2}e^{-1500t} + 16e^{-2400t} \;=\;0$

Multiply by $\displaystyle 2e^{2400t}\!:\;\;2e^{1800t} - 25e^{900t} + 32 \;=\;0$

Let $\displaystyle x \,=\,e^{900t}\!:\;\;2x^2 - 25x + 32 \;=\;0$

Quadratic Formula: .$\displaystyle x \;=\;\frac{25 \pm3\sqrt{41}}{4}$

Back-substitute: .$\displaystyle e^{900t} \;=\;\frac{25\pm3\sqrt{41}}{4} \quad\Rightarrow\quad 900t \;=\;\ln\left(\frac{25\pm3\sqrt{41}}{4}\right)$

. . Therefore: .$\displaystyle t \;=\;\frac{1}{900}\ln\left(\frac{25\pm3\sqrt{41}}{ 4}\right)$

5. very nice as well, thanks. I should've seen this...been out of practice too long I suppose.