# Nakafa Framework: LLM

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

Output docs content for large language models.

---

import { BacterialGrowth } from "@repo/design-system/components/contents/animation-bacterial";

export const metadata = {
  title: "Exponent Concepts",
  description: "Master exponent fundamentals from paper folding to viral spread. Learn base, power definitions, zero/negative/fractional exponents with real applications.",
  authors: [{ name: "Nabil Akbarazzima Fatih" }],
  date: "04/01/2025",
  subject: "Exponents and Logarithms",
};

## From Paper to Pandemic

Have you ever imagined folding a sheet of paper 42 times? If it were possible to do so,
its thickness would exceed the distance from Earth to the Moon! This is because each fold doubles
the thickness of the paper—this is what we call **exponential growth**.

Exponential growth occurs when something increases by a constant multiplier in each
time interval. In early 2020, the world experienced a real example of exponential growth through the
spread of the COVID-19 virus. One infected person could transmit to two people, then four, eight, and so on.

## Definition of Exponents

An exponent is a shorthand way to write repeated multiplication. Imagine you are calculating how many
people are infected with a virus like COVID-19. At each transmission phase, the number of infected people
will increase in an interesting pattern:

<BlockMath math="1 = 2^0 \quad 2 = 2^1 \quad 4 = 2 \times 2 = 2^2 \quad 8 = 2 \times 2 \times 2 = 2^3 \quad 16 = 2 \times 2 \times 2 \times 2 = 2^4" />

This pattern continues, so at phase <InlineMath math="n" />, the number of infected people can be expressed
as <InlineMath math="m(n) = 2^n" />.

For example, if you want to know how many people are infected at phase 5, you just calculate:

<InlineMath math="m(5) = 2^5 = 32" /> people.

## Meaning of Exponent Notation

An expression with an exponent like <InlineMath math="a^n" /> has two important components:

<BlockMath math="a^n" />

Where:

- <InlineMath math="a" /> is the **base** - the number that will be multiplied repeatedly
- <InlineMath math="n" /> is the **exponent** - indicates how many times the base
  is multiplied by itself

In general, if <InlineMath math="a" /> is a real number and <InlineMath math="n" /> is a positive integer, then:

<BlockMath math="a^n = \underbrace{a \times a \times a \times \ldots \times a}_{n \text{ factors}}" />

## Important Definitions in Exponents

Here are some important definitions you need to know:

### Zero Exponent

For any real number <InlineMath math="a" /> where <InlineMath math="a \neq 0" />:

<BlockMath math="a^0 = 1" />

This might seem strange at first, but this definition maintains consistency in the properties of exponents.

### Negative Exponents

For any real number <InlineMath math="a" /> where <InlineMath math="a \neq 0" /> and a positive integer <InlineMath math="n" />:

<BlockMath math="a^{-n} = \left(\frac{1}{a}\right)^n = \frac{1}{a^n}" />

This means a negative exponent equals one divided by the base raised to the same (positive) exponent. This formula is derived from the consistency of exponent properties. To use this formula, you simply flip the base and change the sign of the exponent. Example: <InlineMath math="3^{-2} = \frac{1}{3^2} = \frac{1}{9} = 0.111..." />

### Fractional Exponents

If <InlineMath math="a" /> is a real number where <InlineMath math="a \neq 0" /> and <InlineMath math="n" /> is a positive integer, then:

<BlockMath math="a^{\frac{1}{n}} = p" />{" "}

where <InlineMath math="p" /> is a positive real number such that <InlineMath math="p^n = a" />.

The number <InlineMath math="a^{\frac{1}{n}}" /> is also often called the nth root of <InlineMath math="a" />. This formula emerges as the inverse of exponentiation. To use it, you need to find the number that, when raised to the power of n, will produce a. Example: <InlineMath math="16^{\frac{1}{4}} = 2" /> because <InlineMath math="2^4 = 16" />.

### Mixed Fractional Exponents

If <InlineMath math="a" /> is a real number where <InlineMath math="a \neq 0" /> and <InlineMath math="m,n" /> are positive integers, then:

<BlockMath math="a^{\frac{m}{n}} = \left(a^{\frac{1}{n}}\right)^m" />

This formula is obtained by combining the concepts of roots and exponents. To calculate it, you must first find the nth root of a, then raise it to the power of m. Example: <InlineMath math="8^{\frac{2}{3}} = (8^{\frac{1}{3}})^2 = 2^2 = 4" />.

## Exponential Functions

Exponential functions have the form <InlineMath math="f(x) = a^x" /> where <InlineMath math="a > 0" /> and <InlineMath math="a \neq 1" />. There are two interesting cases:

1. If <InlineMath math="a > 1" />, the function will increase (growth)
2. If <InlineMath math="0 < a < 1" />, the function will decrease (decay)

Exponential functions are very useful in real life because many natural phenomena follow patterns of exponential growth or decay.

## Real-Life Applications

### Bacterial Growth

One bacterium can divide into two, then four, eight, and so on. If <InlineMath math="B_0" /> is the initial number of bacteria and each bacterium divides every hour, then the number of bacteria after <InlineMath math="t" /> hours is:

<BlockMath math="B(t) = B_0 \times 2^t" />

This formula is obtained because the bacterial population doubles at each time interval. The number 2 represents the growth factor.
To use it, multiply the initial amount by 2 raised to the power of the number of intervals that have passed. Example: if there are
initially 100 bacteria and they divide every 30 minutes, after 2 hours (4 intervals) there
will be <InlineMath math="100 \times 2^4 = 100 \times 16 = 1,600" /> bacteria.

<BacterialGrowth />

### Virus Spread

The pattern of virus spread, like COVID-19, also often follows an exponential model, especially in the early phase. If one person can transmit the virus to an average of <InlineMath math="R" /> new people (reproduction number), the number of cases after <InlineMath math="n" /> transmission cycles can be estimated by:

<BlockMath math="C(n) = C_0 \times R^n" />

where <InlineMath math="C_0" /> is the initial number of cases.

This formula is similar to bacterial growth, but with a multiplier factor <InlineMath math="R" /> that can vary. This formula is obtained by multiplying the number of cases by <InlineMath math="R" /> in each transmission cycle. To use it, multiply the initial number of cases by <InlineMath math="R" /> raised to the power of the number of cycles that have passed. Example: if <InlineMath math="R = 2.5" /> and there are 10 initial cases, after 3 transmission cycles there will be <InlineMath math="10 \times 2.5^3 = 10 \times 15.625 = 156.25 \approx 156" /> cases.

### Population Growth

To predict future population numbers, an exponential model can be used with the formula:

<BlockMath math="P(t) = P_0 \times (1 + r)^t" />

where <InlineMath math="P_0" /> is the initial population, <InlineMath math="r" /> is the growth rate, and <InlineMath math="t" /> is time (usually in years).

This formula is obtained by adding the growth percentage <InlineMath math="r" /> to the population at each time interval. The factor <InlineMath math="(1+r)" /> indicates relative growth. To use it, multiply the initial population by <InlineMath math="(1+r)" /> raised to the power of the number of time intervals. Example: if the initial population is 1 million people with 2% growth per year, after 10 years the population becomes <InlineMath math="1,000,000 \times (1 + 0.02)^{10} = 1,000,000 \times 1.22 = 1,220,000" /> people.
