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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

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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

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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

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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 Graph

Exponential Decay Model

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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.

General Graph for the exponential decay:
Graph of Exponential Decay
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