Exponential Decay Formula

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Exponential decay describes a process where a quantity decreases at a rate proportional to its current value. As time increases, the value drops quickly at first and then levels out. Analysts use exponential decay to model processes such as radioactive decay, cooling, and depreciation. It is a specific form of an exponential function. The exponential decay formula is:

N(t) = N₀ × e⁻ʳᵗ

Where:

N(t) is the amount at time t
N₀ is the initial amount
r is the decay rate
e is approximately 2.718

For example, suppose a substance starts with 100 units and decays continuously at a rate of 10% per year. To calculate the amount remaining after 5 years using continuous exponential decay, apply the formula:

N(t) = N₀ × e⁻ʳᵗ

Substitute the known values:

N(5) = 100 × e⁻⁰⋅¹×⁵

First, calculate the exponent:

0.1 × 5 = 0.5

Now:

N(5) = 100 × e⁻⁰⋅⁵

Using the approximate value of e⁻⁰⋅⁵ ≈ 0.6065:

N(5) ≈ 100 × 0.6065 = 60.65

So after 5 years, approximately 60.65 units remain.

N(5) = 100 × e⁻⁰⋅¹×⁵ = 100 × e⁻⁰⋅⁵ ≈ 100 × 0.6065 = 60.65

So only about 60.65 units would remain after 5 years.