*Jennifer Beddoe*Show bio

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

Instructor:
*Jennifer Beddoe*
Show bio

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

The natural base is the base of a natural logarithm and is represented by the letter 'e'. This lesson will define natural base, give some examples of how it is used and conclude with a quiz to test your understanding.

The **natural base**, ** e**, is sometimes referred to as the natural exponent. It is the base for the natural log (

*e* is a constant whose approximate value is 2.71828. It is used extensively in calculating growth and decay problems. It is sometimes called Euler's number after Leonhard Euler, a Swiss mathematician in the 1700s.

Along with 0, 1, π and *I*, *e* is one of the five major mathematical numbers. All of them play important and recurring roles in many areas of math, science and other fields of study.

There are certain general properties that apply to the natural base.

*e*^(*a*+*b*) = (*e*^*a*) + (*e*^*b*)*e*^(*a*/*b*) =*e*^*a*/*e*^*b**e*^(*a*^*b*) =*e*^(*a***b*)

*e*^0 = 1

*e*^(-*a*) = 1/*e*^*a*

*ln*(*e*) = 1

*ln*(*e*^*x*) = *x*

The natural base is used most frequently in calculating exponential growth and decay, whether it be in bacteria, the breakdown of carbon for carbon dating or financials such as investments or stock market gains and losses.

The general equation for growth and decay problems is *A* = *Pe*^(*rt*)

Where:

*A* = ending amount

*P* = initial amount

*e* = natural base = 2.71828

*r* = rate (this number will be negative for decay problems and positive for growth problems.)

*t* = time

Here are some examples of how this equation works.

**1.)** Radium-226, a common isotope of radium, has a half-life of 1620 years and a decay rate of 0.00043. Professor Johnson has a 120-gram sample of radium-226 in his laboratory. How many grams of radium-226 will remain after 200 years?

The first thing we notice about this problem is that it is a decay problem. The equation we use will not change, but the rate will be negative.

When we substitute the values we know into the equation we get:

*A* = 120**e*^(-0.00043*200)

Then simplify

*A* = 120**e*^(-0.086)

*A* = (120)*(0.91759)

*A* = 110 grams of the radium-226 will remain.

**2.)** A certain strain of bacteria is growing on your kitchen counter at a rate of 0.1386 bacteria per minute. Assuming that you start with only one bacterium, how many bacteria could be present after 60 minutes?

This is a growth problem, so the rate will be positive.

Substitute values into the equation to get

*A* = 1**e*^(0.1386*60)

*A* = *e*^8.316

*A* = 4088.7

There will be 4088 bacteria on the counter in an hour.

**3.)** You invest $10,000 at a rate of 5%, compounded monthly. How much money will you have in 4 years?

*A* = 10000**e*^(0.05*48)

NOTE: for this problem, since the rate is in terms of months (compounded monthly), we must also convert the time (4 years) to months (48 months) in order for the equation to work properly.

*A* = 10000**e*^2.4

*A* = 10000*11.02

*A* = $110231.76

The natural base, or *e* is a mathematical constant that is approximately 2.71828. It has great application in both investing and scientific problem solving. *e* is generally thought to be one of the five major numbers in all mathematics.

While working through the lesson, take the time to calculate how well you can:

- Write the natural base
- Recognize general properties that apply to it
- Use the general equation for growth and decay problems

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackRelated Study Materials

- TExES Core Subjects EC-6 (391): Practice & Study Guide
- TExES School Counselor (252): Practice & Study Guide
- Virginia SOL - US History 1865 to Present: Test Prep & Practice
- Virginia SOL - US History to 1865: Test Prep & Practice
- Biology 202L: Anatomy & Physiology II with Lab
- School Counseling and Bullying
- Cultural and Social Diversity in Counseling
- Areas of Clinical Focus
- Fashion and Apparel Design
- Textiles and the Textile Industry
- Role of Student Support in Open & Distance Learning
- TExES Principal Exam Redesign (068 vs. 268)
- Teacher Salary by State
- ESL Resource Guide for Teachers
- What is a Homeschool Co-op?
- How to Start Homeschooling Your Children
- Addressing Cultural Diversity in Distance Learning

- An Ideal Family by Katherine Mansfield: Summary & Characters
- Minoan Civilization: Crafts, Industry & Technology
- Mario Molina: Biography, Facts & Quotes
- Mitochondrial Protein Synthesis
- Justinian & Theodora: Marriage, Facts & Mosaics
- Twelfth Night Act 2 Scene 4: Summary & Analysis
- The Parson's Tale in The Canterbury Tales: Prologue & Summary
- Quiz & Worksheet - Twelfth Night Act 1 Scene 3
- Quiz & Worksheet - Jefferson & the French Revolution
- Quiz & Worksheet - Context for Oedipus Rex
- Quiz & Worksheet - The Canterbury Tales Symbolism & Motif
- Flashcards - Real Estate Marketing Basics
- Flashcards - Promotional Marketing in Real Estate
- Middle School Science Worksheets and Printables
- ESL Games for the Classroom

- Praxis Business Education - Content Knowledge (5101): Practice & Study Guide
- Introduction to Humanities: Certificate Program
- Abnormal Psychology for Teachers: Professional Development
- Communications 101: Public Speaking
- CSET Social Science Subtest I (114): Practice & Study Guide
- Algebra II - Inequalities Review: Homework Help
- Graph Symmetry: Help and Review
- Quiz & Worksheet - Zero-Tolerance Policing
- Quiz & Worksheet - Adding Revenues, Expenses & Dividends with the Accounting Equation
- Quiz & Worksheet - Employee Compensation Accounting
- Quiz & Worksheet - Frequently Used Financial Ratios
- Quiz & Worksheet - Criminal Procedure Rules

- The Six-Step Process for Preparing a Statement of Cash Flows
- Circumference Lesson Plan for Elementary School
- How to Study for Chemistry in College
- New York State Science Standards
- Florida State Standards for Math
- 8th Grade Colorado Science Standards
- Arizona Science Standards for 4th Grade
- WV Next Generation Standards for Math
- What is Asthma Educator Certification?
- Chicago Adult Education
- Romance Writing Prompts
- How Long Does it Take to Learn a Language?

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject