Exponential Growth and Decay

Physical quantities experience changes over time, and they can be studied using exponential growth and decay. When the change is not constant but exponential, it is termed exponential growth or exponential decay.

Exponential growth tells us how something increases with time, and exponential decay shows how something reduces with time.

It is expressed in the form f(x) = abx, where 

a = initial quantity

b = growth factor 

and x represents the time over which the growth or decay occurs.

If the value of b > 1, it represents exponential growth, and if b < 1, it represents exponential decay. 

The concept of exponential growth says that something always grows in relation to its current value. It is similar to the concept of doubling. For example, If a rabbit population doubles every month, the numbers would go 2, then 4, then 8, 16, 32, 64, 128, 256, and so on.

Let us consider a special tree that grows exponentially. Then, 

Its height in mm is ex. 

Where e stands for Euler’s number. 

E = 2.718

Therefore, let us now increase the value of x exponentially and see what happens. 

1-year-old \(e^1\) = 2.7 mm high, which is really tiny.

5-year-old \(e^5 \) = 148 mm high, which is as high as a cup. 

10-year-old \(e^{10}\) = 22 m high, which is as tall as a building.

15-year-old \( e^{15}\) = 3.3 km high, which is as tall as 10 stacked Eiffel towers.

20-year-old \(e^{20}\) = 485 km high, which reaches up into the space.

There is no tree that can grow that tall. Therefore, when someone says that it grows exponentially, we have to think about what they could be meaning. But sometimes, some things can grow exponentially, for a while. 

Therefore, we use a general formula, 

\(y(t) = a × e^{kt}\)

Where y(t) = value at time "t"

a = value at the start

k = rate of growth (when >0) or decay (when <0)

t = time

For example, 

Let us say that 2 months ago, we had 3 mice, and now we have 18. Find the value of the rate of growth. 

Let's start with the formula, 

\(y(t) = a × e^{kt}\)

We know that, 

a = 3, t = 2 and here, y(2) = 18.

Therefore, 

\(18 = 3 × e^{2k}\)

\(6 = e^{2k}\)

Taking the natural logarithm on both sides, 

ln(6) = ln(e2k)

ln(ex)=x, so:ln(6) = 2k

2k = ln(6)

\(k = \frac{ln(6)}{2}\)

Just as some things increase exponentially, some things decay or get smaller exponentially.

For example,

Atmospheric pressure decreases exponentially with increasing altitude.

It decreases by about 12% for every 1000m.

Similarly, the pressure at sea level is approximately 1013 hPa. 

Half-life is the time it takes for a value to decrease to half its original value in exponential decay. We commonly use this concept in radioactive decay, but it also has many other applications. 

For example, if the half-life of caffeine in our body is about 6 hours and if we had one cup of coffee 9 hours ago, how much caffeine is left in our body?

\(y(t) = a × e^{kt}\)

We know that: 

A = 1 cup of coffee

T is in hours and 

For y(6), we have a 50% reduction in caffeine.

Therefore, 

\(0.5 = 1 × e^{6k}\)

Now, let us apply algebra to solve for k.

Let us take the logarithm of both sides.

ln(0.5) = ln(e6k)

ln(ex)=x, so:ln(0.5) = 6k

6k = ln(0.5)

\(k = \frac{ln(0.5)}{6}\)

Therefore, now we can write that, 

\(y(t) = 1 e^{\left(\frac{ln(0.5)}{6}\right)xt}\)

In 6 hours;

\(y(6) = 1 e^{\left(\frac{ln(0.5)}{6}\right)x6} = 0.5\)

Which is correct, as 6 hours is the half-life.

And in 9 hours;

\(y(9) = 1 e^{\left(\frac{ln(0.5)}{6}\right)x9} = 0.35\)

After 9 hours, the amount of caffeine left in our system is about 0.35 of the original amount.

Now, let’s learn how to calculate exponential growth and decay. The formulas to calculate them are given below:
 

Where a or P₀ represents the initial quantity of the substance, r represents the rate of growth, and t represents the time steps. 

Exponential growth and decay describe how quantities increase or decrease rapidly over time in real-world scenarios. Grasping these concepts helps solve problems in science, finance, and everyday life efficiently.
 

• Understand the fromulas correctly and apply them accordingly.
   
• Positive rates indicate growth, while negative rates indicate decay, helping you quickly identify the scenario.
   
• For unknown time or rates in an exponential equation, use logarithms.
   
• Ensure that the rate and time use the same units, such as years, months, or days, to avoid calculation errors.
   
• Apply the formulas to examples like population growth, radioactive decay, or compound interest to strengthen understanding.
   
• Teachers can start teaching exponential growth with meaningful examples. We can make use of real-life situations, such as bacteria doubling, money growing in a bank, increasing social media followers, and rumors spreading.
   
• Similarly, for exponential decay, we can use melting ice, depreciation of a car, the elimination of medicine from the body, and radioactive decay to explain the concept easily.
   
• Parents should use the simplest way to visualize exponential change. Tell them that the growth keeps doubling, and the decay keeps halving.
   
• Teachers can make use of simple numbers to teach children about growth and decay. For example, 2 4816 and 80402010.
   
• Parents can use a folded sheet of paper to demonstrate the effect of doubling the thickness. It provides a simple explanation of growth and decay. 

Exponential growth and decay have many real-life applications. They are used in various fields as mentioned below: 

• Chemistry: In chemistry, we use exponential decay in the concentration of reactants over time to study first-order reactions. This helps us understand the reaction speeds in various chemical processes.

• Biology: Exponential growth and decay are used to analyze viral or bacterial growth. This is an important application because it helps us understand the spread of various diseases. 

• Marketing: While analyzing feedback from customers, the number of responses can grow exponentially. 
   
• Finance: Exponential growth is used to calculate compound interest on savings and investments. It helps predict how money will grow over time with regular interest accumulation.
   
• Physics: Exponential decay models radioactive decay of unstable isotopes. It helps scientists determine half-lives and the remaining quantity of radioactive materials over time.