# Nakafa Learning Content > For AI agents: use [llms.txt](https://nakafa.com/llms.txt) for the site index. Markdown versions are available by appending `.md` to content URLs or sending `Accept: text/markdown`. URL: https://nakafa.com/en/subjects/mathematics/exponential-logarithm/exponential-growth Source: https://raw.githubusercontent.com/nakafaai/nakafa.com/refs/heads/main/packages/contents/material/lesson/mathematics/exponential-logarithm/exponential-growth/en.mdx Explore exponential growth through bacterial multiplication and population dynamics. Learn accelerating functions with worked calculations. --- ## 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: $$f(x) = a^x$$ where $$a > 1$$ Visible text: where Where: - $$a$$ is the base (growth constant) - $$x$$ is the variable (time) Visible text: - is the base (growth constant) - 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 $$30 \text{ bacteria}$$. These bacteria divide into two every $$30 \text{ minutes}$$. Visible text: A researcher observes bacterial growth on a host with an initial condition of . These bacteria divide into two every . If $$x$$ is the growth phase of bacteria every $$30 \text{ minutes}$$, then: Visible text: If is the growth phase of bacteria every , then: | Phase ($$30 \text{ minutes}$$) | $$0$$ | $$1$$ | $$2$$ | $$3$$ | $$4$$ | $$5$$ | | ------------------ | --- | --- | --- | --- | --- | --- | | Number of bacteria | $$30$$ | $$60$$ | $$120$$ | $$240$$ | $$480$$ | $$960$$ | Visible text: | Phase () | | | | | | | | ------------------ | --- | --- | --- | --- | --- | --- | | Number of bacteria | | | | | | | It can be observed that: - For $$x = 0$$, the number of bacteria is $$30$$ - For $$x = 1$$, the number of bacteria is $$60$$ - For $$x = 2$$, the number of bacteria is $$120 = 2^2 \cdot 30$$ - For $$x = 3$$, the number of bacteria is $$240 = 2^3 \cdot 30$$ - For $$x = 4$$, the number of bacteria is $$480 = 2^4 \cdot 30$$ Visible text: - For , the number of bacteria is - For , the number of bacteria is - For , the number of bacteria is - For , the number of bacteria is - For , the number of bacteria is From this pattern, bacterial growth can be modeled with an exponential function: ```math f(x) = 30 \cdot (2^x) ``` ### Visualization of Exponential Growth Graph The graph of the function $$f(x) = 30 \cdot (2^x)$$ shows growth that gets faster as the value of $$x$$ increases. A characteristic feature of exponential growth graphs is their increasingly steep curve. Visible text: The graph of the function shows growth that gets faster as the value of increases. A characteristic feature of exponential growth graphs is their increasingly steep curve. Component: FunctionChart Props: - p: 30 - a: 2 - title: Bacterial Growth - description: Growth of bacteria that divide into two every $$30 \text{ minutes}$$. Visible text: Growth of bacteria that divide into two every . ### Calculating the Number of Bacteria at a Specific Time If we want to calculate the number of bacteria at hour $$5$$, we need to know that hour $$5$$ occurs at phase $$10$$ (because each phase is $$30 \text{ minutes}$$): Visible text: If we want to calculate the number of bacteria at hour , we need to know that hour occurs at phase (because each phase is ): ```math f(10) = 30 \cdot (2^{10}) = 30 \cdot 1024 = 30{,}720 ``` So, at hour $$5$$, there are $$30{,}720 \text{ bacteria}$$. Visible text: So, at hour , there are . ## Variations in Initial Value Exponential growth can have different initial values. In general, if the initial value is $$P_0$$, then the exponential growth function becomes: Visible text: Exponential growth can have different initial values. In general, if the initial value is , then the exponential growth function becomes: ```math f(x) = P_0 \cdot (a^x) ``` For example: - If the initial number of bacteria is $$50$$, then $$f(x) = 50 \cdot (2^x)$$ - If the initial number of bacteria is $$100$$, then $$f(x) = 100 \cdot (2^x)$$ - If the initial number of bacteria is $$200$$, then $$f(x) = 200 \cdot (2^x)$$ Visible text: - If the initial number of bacteria is , then - If the initial number of bacteria is , then - If the initial number of bacteria is , then ### Determining the Initial Value Sometimes we need to determine the initial value $$x_0$$ of an exponential growth if we know its value at a certain time. Visible text: Sometimes we need to determine the initial value 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? Visible text: Suppose we know that the number of bacteria at phase is and the bacteria divide into two at each time interval. What is the initial number of bacteria? We can use the equation $$x_2 = a^2 \cdot x_0$$ with $$a = 2$$: Visible text: We can use the equation with : Component: MathContainer Children: ```math x_2 = a^2 \cdot x_0 ``` ```math 8000 = 2^2 \cdot x_0 ``` ```math 8000 = 4 \cdot x_0 ``` ```math \frac{8000}{4} = x_0 ``` ```math x_0 = 2000 ``` So, the initial number of bacteria is $$2{,}000 \text{ bacteria}$$. Visible text: So, the initial number of bacteria is . ### 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 $$10 \text{ hours}$$: Visible text: 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 : ```math x_{10} = a^{10} \cdot x_0 ``` Substitute the values $$a = 2$$ and $$x_0 = 2{,}000$$: Visible text: Substitute the values and : Component: MathContainer Children: ```math x_{10} = 2^{10} \cdot 2{,}000 ``` ```math x_{10} = 1{,}024 \cdot 2{,}000 ``` ```math x_{10} = 2{,}048{,}000 ``` So, the number of bacteria after $$10 \text{ hours}$$ is $$2{,}048{,}000 \text{ bacteria}$$. Visible text: So, the number of bacteria after is . ## 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. 1. Draw a table and graph showing the growth of these bacteria from phase $$0$$ to phase $$5$$. 2. Model the function that describes the growth of E.coli bacteria every quarter of an hour. 3. Predict how many bacteria will be present after the first $$3 \text{ hours}$$ and $$4 \text{ hours}$$. 2. In $$2015$$, positive HIV-AIDS cases totaled about $$36 \text{ million people}$$. 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 $$2020$$? Visible text: 1. E.coli bacteria cause diarrhea in humans. A researcher observed the growth of of these bacteria on a piece of food and found that these bacteria divide into every quarter of an hour. 1. Draw a table and graph showing the growth of these bacteria from phase to phase . 2. Model the function that describes the growth of E.coli bacteria every quarter of an hour. 3. Predict how many bacteria will be present after the first and . 2. In , positive HIV-AIDS cases totaled about . This number increased by an average of each year from to . If the increase in positive HIV cases in subsequent years is predicted to increase exponentially at each year, how many cases will occur in ? ### Answer Key 1. E.coli bacteria with an initial count of $$50 \text{ bacteria}$$ that divide into two every $$15 \text{ minutes}$$. 1. Bacterial Growth: | Phase ($$15 \text{ minutes}$$) | $$0$$ | $$1$$ | $$2$$ | $$3$$ | $$4$$ | $$5$$ | | ------------------ | --- | --- | --- | --- | --- | ----- | | Number of bacteria | $$50$$ | $$100$$ | $$200$$ | $$400$$ | $$800$$ | $$1{,}600$$ | Growth of bacteria that divide into two every $$15 \text{ minutes}$$. } /> 2. The E.coli bacterial growth function can be modeled as: ```math f(x) = 50 \cdot (2^x) ``` 3. After the first $$3 \text{ hours}$$ means phase $$12$$ (bacteria divide every $$15 \text{ minutes}$$): ```math f(12) = 50 \cdot (2^{12}) ``` ```math f(12) = 50 \cdot 4{,}096 ``` ```math f(12) = 204{,}800 ``` After the first $$4 \text{ hours}$$ means phase $$16$$ (bacteria divide every $$15 \text{ minutes}$$): ```math f(16) = 50 \cdot (2^{16}) ``` ```math f(16) = 50 \cdot 65{,}536 ``` ```math f(16) = 3{,}276{,}800 ``` 2. HIV-AIDS cases: The number of cases in $$2015$$ was $$36{,}000{,}000 \text{ people}$$ with a $$2\%$$ annual increase. Mathematical model: ```math f(x) = 36{,}000{,}000 \cdot (1 + 0.02)^x ``` To calculate cases in $$2020$$ ($$5 \text{ years}$$ after $$2015$$): ```math f(5) = 36{,}000{,}000 \cdot (1.02)^5 ``` ```math f(5) = 36{,}000{,}000 \cdot 1.1040808 ``` ```math f(5) = 39{,}746{,}908 ``` Therefore, the predicted number of HIV-AIDS cases in $$2020$$ is approximately $$39{,}746{,}908 \text{ people}$$. Visible text: 1. E.coli bacteria with an initial count of that divide into two every . 1. Bacterial Growth: | Phase () | | | | | | | | ------------------ | --- | --- | --- | --- | --- | ----- | | Number of bacteria | | | | | | | Growth of bacteria that divide into two every . } /> 2. The E.coli bacterial growth function can be modeled as: 3. After the first means phase (bacteria divide every ): After the first means phase (bacteria divide every ): 2. HIV-AIDS cases: The number of cases in was with a annual increase. Mathematical model: To calculate cases in ( after ): Therefore, the predicted number of HIV-AIDS cases in is approximately . ## 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.