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An Introduction to the Natural Logarithm (ln)

Author: Brink Harrison


Time: 25 minutes
Preparation Time: 5 minutes
Materials:

Calculators with a log function, graph paper
OR
Graphing calculator with ln function
OR
Excel on a computer workstation

Abstract
To encourage students’ comprehension of the dose response principle, an introduction to what the “natural log” (ln) is and how it behaves is recommended. This will help students to create and analyze a dose response graph.


Objectives
Students will be able to:

  • Define a log as the inverse of an exponentials.
  • Explain the benefit of a natural log for biologists.
  • Calculate the natural log of a number.
  • Plot a logarithmic graph.

National Science Education Standards
Math Strand 2 - Data Analysis, Probability, & Discrete Mathematics
Concept 1: Data Analysis (Statistics)

PO 1. Formulate questions to collect data in contextual situations.
PO 2.  Organize collected data into an appropriate graphical representation.
PO 3. Display data as lists, tables, matrices, and plots

Strand 5: Structure & Logic
Concept 1: Algorithms and Algorithmic Thinking

Teacher Background
Logs are the inverse of exponentials.  This relationship can be expressed as:

bx = y is equivalent to logb(y) = x  

  • Pronounced "log-base-b of y equals x"
  • "b" is called "the base of the logarithm”
  • The base b for a logarithm is always positive and not equal to 1
  • The value inside the logarithm is called the "argument" of the log
    A logarithm can be defined with any base, but the most common base is 10 (log on a standard calculator is log base 10)
  • Logarithms with a base e are called the natural logarithm (ln on a calculator)
  • The natural log is commonly used in biology and engineering because of the "natural" properties of the exponential function which enable it to describe growth or decay
  • e is an irrational number, whose decimal value is approximately 2.71828182845904.  


Related and Resource Websites
http://www.purplemath.com/modules/logs.htm
http://www.purplemath.com/modules/logs2.htm
http://www.purplemath.com/modules/logs3.htm http://www.purplemath.com/modules/graphlog.htm http://www.themathpage.com/alg/logarithms.htmhttp://en.wikipedia.org/wiki/Natural_logarithm



 

 


Activity

  1. If you do not have graphing calculators available, distribute graph paper and have students draw a vertical line and horizontal line through the middle of the paper (creating four quadrants). The vertical line will be labeled y, and the horizontal line will be labeled x.
  2. Have students solve for y in the exponential t-chart on “The e Handout”.  Some calculators may have a function for ex = y (Excel does).
  3. Now have students convert ex = y to loge(y) = x for the natural log t-chart.  Before they begin their calculation, point out that x in both t-charts is exactly the same.  Ask them to predict how the graphs may differ.
  4. Have students graph both t-charts on the same graph.  Discuss how they are inverses of one another.
  5. Relate to the dose-response principle.  If dose-response behaves similar to a natural log, what do students expect to happen to an organism’s reaction (response) to a substance as the concentration (dose) increases?  How does the curve change for a very strong substance versus a substance of lesser strength?

-Top-

The e Handout

  e = 2.71828182845904

  1. Solve for “y”

e x = y
Exponential T-chart


x

y

.125

 

.25

 

.5

 

1

 

2

 

4

 

Plot these coordinates on graph paper.

  1. Convert the ex = y to the inverse natural log equation.

 

 

  1. Solve for “y” (Use the ln function on your calculator.)

Natural Log T-chart


x

y

.125

 

.25

 

.5

 

1

 

2

 

4

 

Plot these coordinates on graph paper.  How do you expect this graph to differ from the exponential graph?




Teacher’s Guide – The e Handout

  1. Solve for “y”

e x = y
Exponential T-chart


x

y

.125

1.133148453

.25

1.284025417

.5

1.648721271

1

2.718281828

2

7.389056099

*4

54.59815003

Plot these coordinates on graph paper.

  1. Convert the ex = y to the inverse natural log equation.

 

loge(y) = x

 

  1. Solve for “y” (Use the ln function on your calculator.)

Natural Log T-chart


x

y

.125

-2.079441542

.25

-1.386294361

.5

-0.693147181

1

0

2

0.693147181

*4

1.386294361

Plot these coordinates on graph paper.  How do you expect this graph to differ from the exponential graph?

*Because of limited space on graph paper, the x column ends at 4, but you can make the graphs more dramatic on a graphing calculator if you continue to double values for x; for example, x = 8, x = 16, etc.



Homework

If applicable

Embedded Assessment

 

 


PULSE is a project of the Community Outreach and Education Program of the Southwest Environmental Health Sciences Center and is funded by:


an
NIH/NCRR award #16260-01A1
The Community Outreach and Education Program is part of the Southwest Environmental Health Sciences Center: an NIEHS Award

LOGO - SWEHSC
LOGO - NIEHS Center LOGO - NIEHS

Supported by NIEHS grant # ES06694


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