Exponential equation - how to use ln?

[dragonblood]

Homework Statement

Problem: find t in the following equation

Homework Equations

$$64000e^{-1600t}+4000e^{-400t}=50000e^{-1000t}$$

The Attempt at a Solution

I know the answer: $$t=6.17\cdot\;10^{-4}s$$. But I'm struggling with how to get there. This is my attempt:

Factorizing down to:

$$1000(64e^{-1600t}+4e^{-400t}-50e^{-1000t})=0$$

and further on:

$$1000e^{-400t}(64e^{-1200t}+4-50e^{-600t})=0$$

I realize only what's in the paranthesis is needed, since the 1000e...is never zero:

$$(64e^{-1200t}+4-50e^{-600t})=0$$

I want to ln both sides, but I'm not quite sure how to do that, since there are three parts...

 

[VietDao29]

Homework Statement

Problem: find t in the following equation

Homework Equations

$$64000e^{-1600t}+4000e^{-400t}=50000e^{-1000t}$$

The Attempt at a Solution

I know the answer: $$t=6.17\cdot\;10^{-4}s$$. But I'm struggling with how to get there. This is my attempt:

Factorizing down to:

$$1000(64e^{-1600t}+4e^{-400t}-50e^{-1000t})=0$$

and further on:

$$1000e^{-400t}(64e^{-1200t}+4-50e^{-600t})=0$$

I realize only what's in the paranthesis is needed, since the 1000e...is never zero:

$$(64e^{-1200t}+4-50e^{-600t})=0$$

I want to ln both sides, but I'm not quite sure how to do that, since there are three parts...

Big hint of the day: If you let x = e^-600t. Then what does your last equation become?

 

[dragonblood]

Aha! So i get a simple second degree equation: $$64x^{2}-50x+4=0$$

Thanks :)

But I'm thinking now that these numbers were 'convenient'...are there other ways to solve this if the numbers don't play along this nicely?

 

[VietDao29]

Aha! So i get a simple second degree equation: $$64x^{2}-50x+4=0$$

Thanks :)

But I'm thinking now that these numbers were 'convenient'...are there other ways to solve this if the numbers don't play along this nicely?

Well, I can assure that, you'll rarely meet an inconvenient problem in high school, especially when you are just getting familiar with exponential equations like this.

After some substitution, an exponential equation in general, will become a quadratic equation, or cubic equation, or even a quartic equation (but in special cases, like: ax⁴ + bx² + x = 0, which then can easily be taken down to a quadratic equation by letting t = x²).

So, don't worry. :)