# Nakafa Framework: LLM

URL: /en/subject/high-school/10/mathematics/exponential-logarithm/exponential-growth
Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/subject/high-school/10/mathematics/exponential-logarithm/exponential-growth/en.mdx

Output docs content for large language models.

---

import { FunctionChart } from "@repo/design-system/components/contents/function-chart";

export const metadata = {
  title: "Exponential Growth",
  description: "Explore exponential growth through bacterial multiplication & population dynamics. Learn accelerating functions with step-by-step calculations.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/02/2025",
  subject: "Exponents and Logarithms",
};

## Definition of Exponential Growth Function

Exponential growth is a type of growth where **the rate of change is proportional to its quantity**. In this growth, the value increases more rapidly over time.

The exponential growth function is written as:

<InlineMath math="f(x) = a^x" /> where <InlineMath math="a > 1" />

Where:

- <InlineMath math="a" /> is the base (growth constant)
- <InlineMath math="x" /> is the variable (time)

## Examples of Exponential Growth

Exponential growth is often found in real life, such as the growth of bacteria that divide into two at regular time intervals.

### Bacterial Growth

A researcher observes bacterial growth on a host with an initial condition of 30 bacteria. These bacteria divide into two every 30 minutes.

If <InlineMath math="x" /> is the growth phase of bacteria every 30 minutes, then:

| Phase (30 minutes) | 0   | 1   | 2   | 3   | 4   | 5   |
| ------------------ | --- | --- | --- | --- | --- | --- |
| Number of bacteria | 30  | 60  | 120 | 240 | 480 | 960 |

It can be observed that:

- For <InlineMath math="x = 0" />, the number of bacteria = 30
- For <InlineMath math="x = 1" />, the number of bacteria = 60
- For <InlineMath math="x = 2" />, the number of bacteria = 120 = <InlineMath math="2^2 \cdot 30" />
- For <InlineMath math="x = 3" />, the number of bacteria = 240 = <InlineMath math="2^3 \cdot 30" />
- For <InlineMath math="x = 4" />, the number of bacteria = 480 = <InlineMath math="2^4 \cdot 30" />

From this pattern, bacterial growth can be modeled with an exponential function:

<BlockMath math="f(x) = 30 \cdot (2^x)" />

### Visualization of Exponential Growth Graph

The graph of the function <InlineMath math="f(x) = 30 \cdot (2^x)" /> shows growth that gets faster as the value of <InlineMath math="x" /> increases. A characteristic feature of exponential growth graphs is their increasingly steep curve.

<FunctionChart
  p={30}
  a={2}
  title="Bacterial Growth"
  description="Growth of bacteria that divide into two every 30 minutes."
/>

### Calculating the Number of Bacteria at a Specific Time

If we want to calculate the number of bacteria at the 5th hour, we need to know that the 5th hour occurs at phase 10 (because each phase is 30 minutes):

<BlockMath math="f(10) = 30 \cdot (2^{10}) = 30 \cdot 1024 = 30,720" />

So, at the 5th hour, there are 30,720 bacteria.

## Variations in Initial Value

Exponential growth can have different initial values. In general, if the initial value is <InlineMath math="P_0" />, then the exponential growth function becomes:

<BlockMath math="f(x) = P_0 \cdot (a^x)" />

For example:

- If the initial number of bacteria is 50, then <InlineMath math="f(x) = 50 \cdot (2^x)" />
- If the initial number of bacteria is 100, then <InlineMath math="f(x) = 100 \cdot (2^x)" />
- If the initial number of bacteria is 200, then <InlineMath math="f(x) = 200 \cdot (2^x)" />

### Determining the Initial Value

Sometimes we need to determine the initial value <InlineMath math="x_0" /> of an exponential growth if we know its value at a certain time.

**Example case**:

Suppose we know that the number of bacteria at phase 2 is 8,000 and the bacteria divide into two at each time interval. What is the initial number of bacteria?

We can use the equation <InlineMath math="x_2 = a^2 \cdot x_0" /> with <InlineMath math="a = 2" />:

<div className="flex flex-col gap-4">
  <BlockMath math="x_2 = a^2 \cdot x_0" />
  <BlockMath math="8000 = 2^2 \cdot x_0" />
  <BlockMath math="8000 = 4 \cdot x_0" />
  <BlockMath math="\frac{8000}{4} = x_0" />
  <BlockMath math="x_0 = 2000" />
</div>

So, the initial number of bacteria is 2,000 bacteria.

### Calculating Long-Term Growth

To calculate the number of bacteria after a longer time, we still use the same model. For example, to calculate the number of bacteria after 10 hours:

<BlockMath math="x_{10} = a^{10} \cdot x_0" />

Substitute the values <InlineMath math="a = 2" /> and <InlineMath math="x_0 = 2,000" />:

<div className="flex flex-col gap-4">
  <BlockMath math="x_{10} = 2^{10} \cdot 2,000" />
  <BlockMath math="x_{10} = 1,024 \cdot 2,000" />
  <BlockMath math="x_{10} = 2,048,000" />
</div>

So, the number of bacteria after 10 hours is 2,048,000 bacteria.

## Exercises

1. E.coli bacteria cause diarrhea in humans. A researcher observed the growth of 50 of these bacteria on a piece of food and found that these bacteria divide into 2 every quarter of an hour.

   a. Draw a table and graph showing the growth of these bacteria from phase 0 to phase 5.

   b. Model the function that describes the growth of E.coli bacteria every quarter of an hour.

   c. Predict how many bacteria will be present after the first 3 and 4 hours.

2. In 2015, positive HIV-AIDS cases totaled about 36 million people. This number increased by an average of 2% each year from 2010 to 2015. If the increase in positive HIV cases in subsequent years is predicted to increase exponentially at 2% each year, how many cases will occur in 2020?

### Answer Key

1. E.coli bacteria with an initial count of 50 bacteria that divide into two every 15 minutes.

   a. Bacterial Growth:

   | Phase (15 minutes) | 0   | 1   | 2   | 3   | 4   | 5     |
   | ------------------ | --- | --- | --- | --- | --- | ----- |
   | Number of bacteria | 50  | 100 | 200 | 400 | 800 | 1,600 |

   <FunctionChart
     p={50}
     a={2}
     n={7}
     title="Bacterial Growth"
     description="Growth of bacteria that divide into two every 15 minutes."
   />

   b. The E.coli bacterial growth function can be modeled as:

   <BlockMath math="f(x) = 50 \cdot (2^x)" />

   c. After the first 3 hours means phase 12 (bacteria divide every 15 minutes):

   <div className="flex flex-col gap-4">
     <BlockMath math="f(12) = 50 \cdot (2^{12})" />
     <BlockMath math="f(12) = 50 \cdot 4,096" />
     <BlockMath math="f(12) = 204,800" />
   </div>

   After the first 4 hours means phase 16 (bacteria divide every 15 minutes):

   <div className="flex flex-col gap-4">
     <BlockMath math="f(16) = 50 \cdot (2^{16})" />
     <BlockMath math="f(16) = 50 \cdot 65,536" />
     <BlockMath math="f(16) = 3,276,800" />
   </div>

2. HIV-AIDS cases:

   The number of cases in 2015 was 36,000,000 people with a 2% annual increase.

   Mathematical model:

   <BlockMath math="f(x) = 36,000,000 \cdot (1 + 0.02)^x" />

   To calculate cases in 2020 (5 years after 2015):

   <div className="flex flex-col gap-4">
     <BlockMath math="f(5) = 36,000,000 \cdot (1.02)^5" />
     <BlockMath math="f(5) = 36,000,000 \cdot 1.1040808" />
     <BlockMath math="f(5) = 39,746,908" />
   </div>

   Therefore, the predicted number of HIV-AIDS cases in 2020 is approximately 39,746,908 people.

## Other Applications of Exponential Growth

Exponential growth is also found in other contexts such as:

- Compound interest in investments
- Population growth of living organisms
- Radioactive decay (negative growth)
- Spread of infectious diseases

Exponential growth is very important in various fields such as biology, finance, and physics because it describes many real-world phenomena.
