Math Help! Word problems with compounds and interest rates.?

[EDIT] I didn't notice the "how to continue from there" bit until now.

The wording of the question is confusing. Po is the original amount, where P(t) gives you the amount after whatever mount of time, t.

P = Po e^(-0.00012 t)

P/Po = e^(-0.00012 t)

Oh cool! We know P/Po is some percentage of the original amount, which they totally gave us. So just solve for t from there.

0.12 = e^(-0.00012 t)

To undo e we use ln

ln(0.12) = ln(e^(-0.00012 t))

ln(0.12) = -0.00012 t

t = -ln(0.12)/0.00012

For the second part you are just setting d(t) = 2 d(0) and solving for t.

I should probably elaborate on that a bit since you my find this to be something you find yourself using in the real world. Doubling time means the time it takes for the initial value to double. Initial as in t=0, and double as in twice whatever the function yields at t = 0.

So lets do the first part, first

Given

d(t) = 1.83(1.9)^t

Remember the function produces money in billion dollar units! so put 10, not 10,000,000,000.

10 = 1.83(1.9)^t

10/1.83 = 1.9^t

We can use ln properties to get t separated

ln(10/1.83) = ln(1.9^t)

ln(10/1.83) = t ln(1.9)

t = ln(10/1.83)/ln(1.9)

Now the doubling time business

d(0) = 1.83(1.9)^0 = 1.83

2 d(0) = 2*1.83 = 3.66

3.66 = 1.83(1.9)^t

3.66/1.83 = (1.9)^t

2 = 1.9^t

ln(2)/ln(1.9) = t (I am skipping steps since you know how to do that much by now :) )

As a hint: doubling time always works out like it did above. You end up getting ln(2)/ln(base) = t, where base is whatever exponent was attached to the t. It is not worth memorizing a formula, just know the process and definitions.