Decay Rate of Cs Time

Feb 2018
Here is the question give to me by my teacher:
A sample of 137Cs (half-life = 30.1 years) decays at a rate of 10^10 decays per second. How long will it take for the decay rate to decrease to 10^7 decays per second?

The work I have done so far is:
dN(t=0) = -λNo, where dN(t) = Noe^(-λt)
(Although online a website as it as Noe -λt, so not sure if λt should be raised or not)

Because I am trying to find the difference I can make this formula equal to itself:
Noe^(-λt) = Noe^(-λt)
λ is my decay constant (10^10 and 10^7)
e is e (2.71828)
t is the time (guessing at 0 and x (what the question is asking for))
No is the amount - which I do not know

Overall, the part I am confused with is if this is even correct, and if so how do I find the No of the amounts?
Apr 2017
I don't like using formulae and equations unless absolutely necessary, they tend to stop the thinking process ...Better derive the equation myself from first principles ,

The question is how long will it take for the decay rate to reduce a thousand fold ...

When the Half life 30.1 years

after one half life it's a half....2HL it's a 1/4 .....3 HL it's an 1/8 ..... 4 HL it's a 1/16th

So if X is the number of half lifes then the radiation will be One over 2 to the power X

2 to the power 10 is 1024 ... but we want the number which will give a 1,000 should be just less then 10.

Using my calculator I get an exact value of 9.9658 half lives = Almost exactly 300 years ( 299.97)
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