Chapter 1: An Introduction to Exponents and Logarithms

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Contributor:

Erica Madebeykin
PGDip Researcher

Creation Date: October 20, 2024

❤️ Reference Acknowledgments: Duke University (world rank#27)

This review reflects the author's learning and thoughts based on the materials covered in the FREE online course provided by the Duke University, delivered on the coursera platform.
A heartfelt gratitude to Professors Daniel Egger and Paul Bendich for delivering such an amazing foundational math course!

References

1. Biology LibreTexts. (2023). 45.2A: Exponential Population Growth. Retrieved from https://bio.libretexts.org/Bookshelves/Introductory_and_General_Biology/General_Biology_(Boundless)/45:_Population_and_Community_Ecology/45.02:_Environmental_Limits_to_Population_Growth/45.2A:_Exponential_Population_Growth

2. Cormen, T. H., Leiserson, C. E., Rivest, R. L., & Stein, C. (2009). Introduction to Algorithms (3rd ed.). MIT Press.

3. Egger, D., & Bendich, P. (2020). Data Science Math Skills. Duke University, Coursera. https://www.coursera.org/learn/datasciencemathskills

4. Mankiw, N. G. (2021). Principles of Economics (9th ed.). Cengage Learning.

5. PhoenixNAP. (2024). Gigabyte Definition - History, Usage, Storage. Retrieved from https://phoenixnap.com/glossary/gigabyte-definition

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Table of Contents

• 1.1 The Importance of Exponents in Real Life
• 1.1.1 Using Exponents in Scientific Notation
• 1.1.2 Other Practical Applications of Exponents
  • A. Population Growth
  • B. Financial Calculations
  • C. Computer Science
• 1.2 Exponents Fundamentals: A Comprehensive Guide
• 1.2.1 Positive Exponents and its 3 Properties
  • 3 Properties of Positive Exponents:
• 1.2.2 Negative Exponents and its 2 properties
  • Property #1 The Negative Exponent Rules: Converting negative exponents to Fraction and Decimal formats
  • Property #2: Multiplication of Powers with Negative Exponents
• Notes on Working with One and Two Exponents
• 1.2.3 Zero as Exponent
  • Properties of Zero Exponent:
• 1.2.4 Fractional Exponents and its 3 Properties
  • Understanding Fractional Exponents
  • 3 Key Properties of Fractional Exponents
  • Fractional Exponent Property #1: Root and Power Relationship
  • Fractional Exponent Property #2: Multiplication of Fractional Exponents
  • Important Note: Special Cases with Fractional Exponents
  • Fractional Exponent Property #3 : Quotient Property
• 1.3 Converting Exponents to Logarithms

1.1 The Importance of Exponents in Real Life

What is Exponentiation?

Exponentiation is the process of raising a number (the base) to a power (the exponent).

For example, in the expression (3^3), the base is 3, and the exponent is 3. This means (3) is multiplied by itself three times:

$$3^3=3\times 3\times 3=27$$

Exponents have various practical uses in real life, such as in scientific notation, finance, computer science, and much more.

1.1.1 Using Exponents in Scientific Notation

Exponents play a main role in scientific notation, a convenient method to express very large or very small numbers. It's typically written in the form

$(a\times {10}^n)$, where a is a number between 1 and 10, and n is an integer that indicates the power of ten.

Positive exponents indicate large values by multiplying the base number by powers of ten, while negative exponents indicate small values by dividing the base number by powers of ten. They both play a big role in scientific notation, especially in fields like physics, biology, and chemistry.

Understanding these concepts helps in grasping how numbers can represent extremely large or small quantities in various scientific contexts.

Scientific Notation Example: Mass of Earth

• Positive exponent: the mass of the Earth can be expressed in scientific notation as:

$5.97\times {10}^{24}\text{ kg}$

This means:

• The base number is 5.97.
• The positive exponent is 24, which indicates that we multiply 5.97 by 10 twenty-four times.

In standard form without scientific notation, this can be written as:

Without Scientific Notation : Writing Mass of Earth in kg

$$5.97\times {10}^{24}$$$$=5.97\times 1,000,000,000,000,000,000,000,000$$$$=5,970,000,000,000,000,000,000,000\text{ kg}$$

This represents a very large number (approximately 5.97 septillion kilograms), such as the mass of the Earth.

When we express a number with a negative exponent, it indicates that the base number is divided by ten raised to the absolute value of that exponent.

The Size of E. coli Bacteria Represented with Negative Exponents in Scientific Notation

• Negative exponent: the bacteria Escherichia coli (E. coli) is about 1 to 2 micrometers in length, which can be expressed as:

$1.00\text{to}2.00\times {10}^{-6}\text{m}$ where -6 exponent indicates a negative exponent.

This means:

• The base number is between 1.00 and 2.00.
• The exponent is -6, which indicates that we divide by 10 six times.

In standard form without scientific notation, this can be expressed as:

E. coli Bacteria size Without Scientific Notation

$$1.00\times {10}^{-6}=\frac{1.00}{1,000,000}=0.000001\text{ m}$$$$\begin{gathered} 2.00\times {10}^{-6}=\frac{2.00}{1,000,000} \\ =0.000002\text{ m} \end{gathered}$$

This represents a very small range of lengths (from 0.000001 meters to 0.000002 meters), which is approximately the size of bacteria like Escherichia coli (E. coli).

1.1.2 Other Practical Applications of Exponents

Exponents are commonly used in various fields such as mathematics, science, and engineering to represent exponential growth, calculate areas, and solve equations.

A. Population Growth

In biology, population growth can often be modeled using exponential functions. For example, if a population doubles every year, the growth can be represented as:

$$P(t)=P_0\times 2^t$$

where (P_0) is the initial population and (t) is time in years (Biology LibreTexts, 2023).

B. Financial Calculations

In finance, compound interest is calculated using exponents. The formula for compound interest is:

$$A=P(1+r{)}^n$$

where:

• (A) is the amount of money accumulated after (n) years, including interest.
• (P) is the principal amount (the initial sum of money).
• (r) is the annual interest rate (decimal).
• (n) is the number of years the money is invested or borrowed (Mankiw, 2021).

C. Computer Science

In computer science, exponents are used to describe algorithm complexity and data storage capacities. For instance, a binary system uses powers of two:

A byte consists of 2^8 or 256 possible values.

$$2^8=256$$

The complexity of certain algorithms can be expressed as

$$O(n^k)$$

where (k) indicates how the performance scales with input size (Cormen et al., 2009).

Exponents are also essential in describing data storage capacities (megabytes, gigabytes) and processing speeds (megahertz). For example, a gigabyte is defined as ${10}^9$ bytes, highlighting the exponential nature of data growth in technology (PhoenixNAP, 2024).

1.2 Exponents Fundamentals: A Comprehensive Guide

Exponents are usually Integers that can be Positive, Negative, or Zero, each having distinct properties and applications (Egger & Bendich, 2020).

On the other hand, exponents can be also expressed as Fractions in equations.

1.2.1 Positive Exponents and its 3 Properties

Key Pointers:

• A positive exponent indicates that a base number is multiplied by itself a specified number of times.
• For example, ( a^n ) means that ( a ) is multiplied by itself ( n ) times.

Examples:

• $2^3=2\times 2\times 2=8$
• $5^4=5\times 5\times 5\times 5=625$

3 Properties of Positive Exponents:

Positive Exponents 3 Properties

• Product of Powers:

  $$a^m\times a^n=a^{m+n}$$

• Power of a Power:

  $$(a^m{)}^n=a^{m\cdot n}$$

• Power of a Product:

  $$(ab{)}^n=a^n\times b^n$$

Property #1: Product of Powers

When multiplying two expressions with the same base, you can add the exponents:

$$a^m\times a^n=a^{m+n}$$

Example:

$$2^3\times 2^2=2^{3+2}=2^5=32$$

Property #2: Power of a Power

When raising an exponent to another exponent, you can multiply the exponents:

$$(a^m{)}^n=a^{m\cdot n}.$$

Example:

$$(3^2{)}^4=3^{2\cdot 4}=3^8=6561$$

Property #3: Power of a Product

When raising a product to an exponent, you can distribute the exponent to each factor:

$$(ab{)}^n=a^n\times b^n.$$

Example:

$$(2\cdot 3{)}^3=2^3\cdot 3^3=8\cdot 27=216$$

1.2.2 Negative Exponents and its 2 properties

Negative Exponents have two properties:

Two Properties of Negative Exponents:

• Negative Exponent Rule: $a^{-n}=\frac{1}{a^n}$
• Multiplication of Powers: $a^m\times a^{-n}=\frac{a^m}{a^n}=a^{m-n}$

Property #1 The Negative Exponent Rules: Converting negative exponents to Fraction and Decimal formats

Key Pointers:
A negative exponent also represents a fraction, where the base is in the denominator:

Converting Negative Exponent to fraction

$$a^{-n}=\frac{1}{a^n}$$

So, given the example:
The bacterium Escherichia coli (E. coli) is approximately 1 to 2 micrometers in length. The negative exponent means that the base (10 in this case) is in the denominator. We can rewrite the negative exponent ${10}^{-6}$ as a fraction:

Negative exponent in Fraction Format

$${10}^{-6}=\frac{1}{{10}^6}=\frac{1}{1,000,000}$$

Using the same example ${10}^{-6}$, we can also transform the negative exponent into decimal form:

Negative Exponent in Decimal Format

• To understand this as a decimal, we first convert it to a fraction, then to decimal:

$${10}^{-6}=\frac{1}{1,000,000}=0.000001$$

Property #2: Multiplication of Powers with Negative Exponents

Multiplying Powers with Negative Exponents

• Multiplication of Powers:
• One negative exponent

$$a^m\times a^{-n}=\frac{a^m}{a^n}=a^{m-n}$$

• Two negative exponents

$$a^{-m}\times a^{-n}=\frac{a^{-m}}{a^{-n}}=a^{-(m+n)}$$

Notes on Working with One and Two Exponents

One Negative Exponent

This can result to either negative or positive result**

• Negative Exponent as Result

$$a^3\times a^{-5}=a^{3+(-5)}=a^{3-5}=a^{-2}=\frac{1}{a^2}$$

is also the same as

$$a^3\times a^{-5}=a^3\times \frac{1}{a^5}=\frac{a^3}{a^5}=a^{3-5}=a^{-2}=\frac{1}{a^2}$$

• Positive Exponent as Result

$$a^5\times a^{-3}=a^{5+(-3)}=a^{5-3}=a^2$$

Two Negative Exponents will always result in negative exponent or fraction

1. Using the property of exponents:

   $$a^{-3}\times a^{-5}=a^{-3+(-5)}=a^{-8}=\frac{1}{a^8}$$

2. Using the definition of negative exponents:

   $$a^{-3}\times a^{-5}=\frac{1}{a^3}\times \frac{1}{a^5}=\frac{1}{a^{3+5}}=\frac{1}{a^8}$$

1.2.3 Zero as Exponent

Properties of Zero Exponent:

Key Pointers

• Zero Exponent Rule: Any non-zero base raised to the power of zero is equal to one: $a^0=1$ (for $a\ne 0$).
• In the case that the base is zero and is raised to the exponent of zero, it will be undefined.

Examples:

• $3^0=1$
• ${10}^0=1$
• $0^0=undefined$

1.2.4 Fractional Exponents and its 3 Properties

Understanding the fractional exponents is essential in algebra, as it shows the relationship between powers and roots. Mastering how to manipulate and apply these exponents can significantly strengthen your foundational skills for higher-level mathematics, such as calculus and statistics.

Whether you're simplifying expressions or solving equations, mastering fractional exponents will serve you well!

Understanding Fractional Exponents

Fractional exponents are a powerful tool in mathematics that combine the concepts of exponents and roots. The basic formula for a fractional exponent is expressed as:

Fractional Exponents

$$x^{\frac{a}{b}}=\sqrt[b]{x^a}$$

• This means that when you raise a number x to a fractional exponent a/b.

• It is equivalent to taking the b-th root of x raised to the power of a (Egger & Bendich, 2020).

3 Key Properties of Fractional Exponents

Key Properties of Fractional Exponents

1. Root and Power Relationship: The numerator of the fractional exponent indicates the power to which the base is raised, while the denominator indicates the root to be taken (Egger & Bendich, 2020).

   Formula:

   $$x^{\frac{a}{b}}=\sqrt[b]{x^a}$$

2. Multiplication of Fractional Exponents: When multiplying numbers with fractional exponents, you can add the fractions:

   $$x^{\frac{a}{b}}\times x^{\frac{c}{d}}=x^{\frac{ad+bc}{bd}}$$

3. Quotient Property of Fractional Exponents: When dividing two expressions with the same base, you can subtract the exponent in the denominator from the exponent in the numerator:

   $$\frac{x^{\frac{m}{n}}}{x^{\frac{p}{q}}}=x^{\frac{m}{n}-\frac{p}{q}}=x^{\frac{mq-pn}{nq}}$$

Fractional Exponent Property #1: Root and Power Relationship

Root and Power Relationship

To illustrate this property, consider the expression

$$8^{\frac{2}{3}}$$

According to our formula, we can rewrite it as:

$$8^{\frac{2}{3}}=\sqrt{8^2}$$

Calculating this step-by-step:

• First we solve the numerator of the factorial exponent 2/3, by raising the base 8 to the power of 2. This results to 64.

$$8^2=64$$

• Then for the denominator of the fractional exponent which is 3 and equivalent to "cube root", so we find the cube root of 64:

$$\sqrt[3]{64}=4$$

Therefore, we conclude that:

$$8^{\frac{2}{3}}=4$$

Fractional Exponent Property #2: Multiplication of Fractional Exponents

When multiplying two numbers with fractional exponents, you can combine them by adding their exponents.

Multiplication of Fractional Exponents example:

$$x^{\frac{1}{2}}\times x^{\frac{1}{3}}=x^{\frac{1}{2}+\frac{1}{3}}$$

To add these fractions, find a common denominator (which is 6):

$$\frac{1}{2}=\frac{3}{6},\text{and}\frac{1}{3}=\frac{2}{6}$$

So, we have:

$$x^{\frac{1}{2}+\frac{1}{3}}=x^{\frac{3+2}{6}}=x^{\frac{5}{6}}$$

Important Note: Special Cases with Fractional Exponents

Special Cases with Fractional Exponents

• Zero Numerator: If the numerator of the fractional exponent is zero, then regardless of the base (as long as it’s not zero), the result will always be one:

  $$x^{\frac{0}{b}}=1(x\ne 0)$$

• Negative Base with Even Roots: If you have a negative base and an even denominator in your fractional exponent, the result will be undefined in real numbers:

  $$(-4{)}^{\frac{1}{2}}=\sqrt{-4}(\text{undefined in real numbers})$$

Fractional Exponent Property #3 : Quotient Property

Yes, fractional exponents do have a quotient property. The quotient property for exponents states that when you divide two expressions with the same base, you can subtract the exponent in the denominator from the exponent in the numerator. This applies to fractional exponents as well.

Quotient Property of Exponents

The quotient property can be expressed as:

$$\frac{a^m}{a^n}=a^{m-n}$$

This means that if you have a base a raised to the power m in the numerator and the same base a raised to the power n in the denominator, you can simplify it by subtracting the exponent in the denominator from the exponent in the numerator.

Application with Fractional Exponents

When dealing with fractional exponents, this property still holds.

For example, if you have:

$$\frac{x^{\frac{3}{2}}}{x^{\frac{1}{2}}}$$

Using the quotient property, you can simplify it as follows:

$$x^{\frac{3}{2}-\frac{1}{2}}=x^{\frac{2}{2}}=x^1=x$$

This demonstrates that fractional exponents also adhere to the quotient property, allowing for simplification just like integer exponents do.

1.3 Converting Exponents to Logarithms

Understanding the relationship between exponents and logarithms is essential in algebra. Logarithms provide a way to express exponentiation in a different form.

Important concept:

Exponents and logarithms are inverse operations, meaning that they undo each other.

Definition of Logarithm:

The logarithm is the inverse operation of exponentiation. If you have an equation of the form:

$$b^y=x,$$

then the logarithm can be expressed as:

$$y={\log }_{b}x$$

• We can read this as "logarithm to the base b of x is y"".
• Here, b is the base, y is the exponent, and x is the result of the exponentiation.

Examples of Converting Exponents to Logarithms

Example 1: Convert this exponential equation to logarithmic form:

$$2^3=8$$

• The equivalent logarithmic form is:

$$3={\log }_{2}8$$

Example 2: Convert this exponential equation to logarithmic form.

$${10}^4=10000$$

• The equivalent logarithmic form is:

$$4={\log }_{10}10000$$

With these foundational concepts in mind, we can now explore logarithms in greater detail in the next chapter. In Chapter 2, we will examine logarithmic functions, their properties, and applications.