How to Solve Exponential Growth Problems Easily
Before answering what is exponential growth? Let us know, why is it called exponent? An exponent refers to the number of times a number is multiplied to itself. We can have a clear idea on the topic by solving exponential growth equations.
Something which always grows in relation to the current value is known as exponential growth. Exponential growth is also known as doubling the existing number. Let us take an example: If the population of rabbits grows every month, then we would have 2, then 4, 8, 16, 32, 64, 128, 256, and further carried on. The formula used in solving exponential growth equations is y = abx.
Exponential Growth Graph
The general shape of an exponential growth graph is ‘J’ shape. This shows that the graph is always increasing in nature. An exponential growth graph is drawn using the function of the form y = abx where, a > 0 and b > 1. Note that the graph shoots upward rapidly as ‘X’ increases. This is because of the doubling behavior of the exponential. Below given graph is an example of exponential growth.
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Exponential Growth Equation
For solving exponential growth equations you need to have equations with an exponential expression on both sides of the equal sign. Then compare the powers and solve the given equation. Let us solve one basic problem to understand it better.
Q. Solve 51-x = 54
As the bases are the same we can equate the powers and solve the problem.
1 - x = 4
x = -3
This is the required solution to the above problem.
It is used in solving exponential growth equations when there is a comparison between two cases.
Exponential Decay
Something which ‘Decay’(gets smaller) with respect to time exponentially is known as exponential decay. Exponential decay is applied when anything decreases compared to the current value exponentially. For example, we can take atmospheric pressure of the air around us which decreases as we go up. It decreases by about 12% for every 1000m which is exponential decay.
Exponential Decay Formula
The formula used to express the exponential decay is y = a(1 - b)x.
Where y is the final amount, a is the original amount, b is the decay factor, and x the amount of time that has passed. Exponential decay formula is useful in solving a variety of real-world problems, Most popular application is to track inventory that is used regularly in the same quantities.
Solved Problems
Q. The population of a city in 2016 estimated to be 35,000 people with an annual rate of increase of 2.4%.
Solution: Now let us find the growth factor,
After one year population would be = 35,000 + 0.024(35,000)
By factoring above equation becomes = 35,000 (1 + 0.024)
The growth factor is b = 1.024 (Remember that it is greater than 1)
Now, The general formula for exponential growth is y = abx
Substituting the value in above formula y = 35,000(1.024)x
Consider, that we are using this estimate of the population in 2020 to the nearest hundred people.
Y = 35,000 (1.024)4≈ 38,482.91 ≈ 38,500
So, the estimated population of the city in 2020 is 38,500.
FAQs on What Is Exponential Growth in Maths?
1. What is exponential growth in Maths?
In mathematics, exponential growth describes a process where a quantity increases at a rate proportional to its current value. Unlike linear growth, which adds a constant amount over time, exponential growth multiplies by a constant factor. This results in an extremely rapid increase, where the growth rate itself accelerates as the quantity gets larger, often represented by a J-shaped curve on a graph.
2. What is the mathematical formula used to calculate exponential growth?
The primary formula to model discrete exponential growth is:
y(t) = a(1 + r)t
Where:
- y(t) is the final amount after time t.
- a is the initial amount.
- r is the growth rate (expressed as a decimal).
- t is the number of time intervals.
3. What is the main difference between exponential growth and linear growth?
The main difference lies in how the quantity increases. Linear growth involves adding a constant amount in each time period, resulting in a straight-line graph. For example, saving ₹100 every month. In contrast, exponential growth involves multiplying by a constant factor or percentage, causing the growth to speed up over time. This creates a steep, upward-curving graph, like an investment that grows by 10% each year.
4. What are some real-world examples of exponential growth?
Exponential growth is observed in many real-world phenomena. Some common examples include:
- Compound Interest: Money in a savings account where interest is earned on both the principal and the accumulated interest.
- Population Growth: Under ideal conditions with unlimited resources, a population of bacteria, animals, or humans can grow exponentially.
- Spread of Viruses: During the initial phase of an epidemic, the number of infected individuals often grows exponentially.
- Viral Content on Social Media: The number of shares or views of a popular video can increase exponentially as more people share it with their followers.
5. How can you identify exponential growth from a graph?
A graph showing exponential growth has a distinct shape known as a J-curve. It starts off growing slowly, appearing almost flat or horizontal. As the x-value (time) increases, the curve bends upwards and becomes progressively steeper, indicating that the rate of growth is accelerating. It is a non-linear curve that increases sharply without bound.
6. What is exponential decay, and how does it differ from exponential growth?
Exponential decay is the opposite of exponential growth. It describes a process where a quantity decreases at a rate proportional to its current value. The key difference is in the growth factor. In the formula y = a(b)t, exponential growth occurs when the base b > 1 (e.g., a 10% increase means b=1.10). Exponential decay occurs when 0 < b < 1 (e.g., a 10% decrease means b=0.90). Common examples of decay include radioactive decay of elements and depreciation of a car's value.
7. Why is Euler's number 'e' important for understanding exponential growth?
Euler's number, 'e' (approximately 2.718), is the base for natural or continuous growth. It represents the maximum possible, perfectly compounded growth rate. While the formula a(1 + r)t models growth in discrete steps (like yearly interest), the formula Pert models growth that is happening constantly at every instant. This makes 'e' fundamental for describing natural phenomena in physics, biology, and finance that grow continuously, not just at fixed intervals.
8. Can a quantity grow exponentially forever in the real world? What are the limitations?
No, in the real world, exponential growth cannot continue indefinitely. Mathematical models assume ideal conditions, but physical environments have limitations. These limiting factors prevent unchecked growth and include:
- Resource Scarcity: Limited food, water, or space for a growing population.
- Carrying Capacity: The maximum population size an environment can sustain.
- Competition: Increased competition for resources among individuals.
- Predation and Disease: As a population grows, it becomes more susceptible to predators and the spread of diseases.
