Çözüldü Exponential Decay - Differential Equation - Logarithm

A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where N is the quantity and λ is a positive rate called the exponential decay constant (disintegration constant, rate constant, or transformation constant): dN(t) / dt = -λ·t and the half-life of the matter equals ln(2) / λ, where N(t) is the quantity at time t, and N₀ = N(0) is the initial quantity, that is, the quantity at time t = 0.
Considering the aforementioned information,
If there are 10^12 atoms in a radioactive matter of which its half-life is thirty days, how many days does it approximately take until there are 10^4 atoms left? [log(e) ≈ 0.434, ln(2) ≈ 0.693]

A) 405
B) 605
C) 805
D) 1005
E) 1205

Integrating (homework for the interested students) both sides of the equation dN(t) / dt = -λ·t yields N(t) = N₀·e^(-λ·t) where N₀ is the initial amount of the matter.
30 = ln(2) / λ ⇒ λ = ln(2) / 30
10^4 = 10^12·e^[ -t·ln(2) / 30 ]
10^(-8) = e^[ -t·ln(2) / 30 ]
10^8 = e^[ t·ln(2) / 30 ]
8·log(10) = [ t·ln(2) / 30 ]·log(e)
8·1 = t·ln(2)·log(e) / 30
8·30 / [ ln(2)·log(e) ] = t
t ≈ 240 / (0.693·0.43)
t ≈ 805.4 days.

Source: https://en.wikipedia.org/wiki/Exponential_decay