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Exponential Decay

Exponential decay occurs when some quantity regularly decreases by a fixed percentage. Exponential decay is said to occur when there is a decrease in the value of a quantity, and in which the rate of decay is directly proportional to value of the quantity. The graph of the exponential function, $y$ = $abx$, will be decaying when $a > 0$ and $b$ is lies between $0$ and $1$.

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Exponential Decay Definition

Exponential decay is decrease of the quantity at a rate proportional to its value. If a quantity $N$ decreases at a rate proportional to the amount present at time $t$, then the quantity can be written as $N(t)$ = $N_0e^{-kt}$, where $N_0$ is the value of $N$ at time $t$ = $0$ and $N$ decreases as $t$ increases and $k$ is called an exponential decay function.

Exponential Decay Formula

As the term, exponential decay, itself suggest it is the study of decay and thus the reverse of exponential growth.

Mathematically, Rate of change of $N(t)$ = $-\ (constant)\ \times\ N(t)$ or

$\frac{dN}{dt}$ = $-\ kN$

$\frac{dN(t)}{N(t)}$
= $-\ k\ d(t)$

Integrating the above equation,

ln $N(t)$ = $-\ kt\ +\ C$

Here, $C$ is the constant of integration.

The above equation can be written as,

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

Now, at $t$ = $0$ (initial condition) $N\ (0)$ = $N_0$

Here, $N_0$ is the initial value or starting value of quantity $N$.

So the expression for exponential decay can be written as,

$N(t)$ = $N_0e^{-kt}$.

Exponential decay formula can be written as:

$N(t)$ = $N_0e^{-kt}$

Exponential Decay Function

Exponential decay is the change that occurs when an original amount is decreases by a consistent rate over time.

Example:

In a lab experiment $1000$ bacteria were kept in a certain environment. After two days, the number of bacteria was reduced to $800$. How many bacterium will there be $7$ days after the initial count of $1000$ bacteria?

Solution:

Given data,

Initial number of bacteria, $N_0$ = $1000$

Number of bacteria after $2$ days,

$N$ = $800$

This problem can be solved in two steps.

First step is to find the decay constant K and the second step is to find the number of bacteria.

Decay Constant $N(t)$ = $N_0\ e^{-kt}$
Or
We can write the above equation in the logarithmic form as,

ln $( N / N_0 )$ = kt ln $(\frac{800}{1000})$

= $2.k$

$k$ = - $0.112$ per day

So, the decay constant is $-0.112$ per day

Now, we have to find the number of bacteria after $7$ days,

$N$ = $N_0\ e^{-kt}$

Substituting the values, $N$ = $1000\ \times\ e^{-0.112\ \times\ 7}$

$N$ = $457$

Therefore, the number of bacteria after $7$ days will be $457$.

Exponential Decay Model

The model for exponential growth or decay is given by, $N(t)$ = $N_0\ e^{kt}$, where $t$ is time, $N_0$ is the origin amount of the quantity and $N(t)$ is the amount after $t$. The number $k$ is the constant that is determined by the rate of the growth or decay. If $k < 0$, it represents exponential decay.