To solve an exponential decay problem, you can utilize the formula given by

$N(t) = N_0 e^{-kt}$

Exponential decay describes the process by which a quantity decreases over time at a rate that is proportional to its current value. This formula, $N(t) = N_0 e^{-kt}$, allows us to determine the remaining amount $N(t)$ after a specific time $t$. In this expression, $N_0$ represents the initial quantity, $k$ is the decay constant, and $e$ is the base of the natural logarithm, approximately equal to $2.718$.

To begin, identify the initial quantity $N_0$ and the decay constant $k$. The decay constant $k$ is typically provided in the problem statement or can be derived from the available data. If you know the half-life of the substance (the time required for the quantity to reduce to half its initial value), you can calculate $k$ using the formula

$k = \frac{\ln(2)}{\text{half-life}}$

Next, substitute the known values into the exponential decay formula. For example, suppose you start with $100$ grams of a substance that has a decay constant of $0.03$ per year, and you want to find out how much remains after $5$ years. In this case, you would set $N_0 = 100$, $k = 0.03$, and $t = 5$. Plugging these values into the formula gives:

$N(5) = 100 e^{-0.03 \times 5}$

Finally, perform the calculation using a calculator. For our example, we find:

$N(5) = 100 e^{-0.15} \approx 100 \times 0.8607 \approx 86.07$

Thus, after $5$ years, approximately $86.07$ grams of the substance will remain.