An introduction to the Natural Log (ln)
Time:
25 minutes
Preparation
5 minutes
Time:
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:
i.
Define a log as the inverse of an exponentials.
ii.
Explain the benefit of a natural log for biologists.
iii.
Calculate the natural log of a number.
iv.
Plot a logarithmic graph.
Arizona Math Education Standards:
Math Strand 2 - Data Analysis, Probability, & Discrete Mathematics
Concept 1: Data Analysis (Statistics)
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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
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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)
http://pulse.pharmacy.arizona.edu/10th_grade/disease_epidemics/science/medic.html
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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.
Additional Resources
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.htm
http://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?
http://pulse.pharmacy.arizona.edu/10th_grade/disease_epidemics/science/medic.html
The e Handout
e = 2.71828182845904
1. Solve for “y”
ex = y
Exponential T-chart
x
y
.125
.25
.5
1
2
4
Plot these coordinates on graph paper.
2. Convert the ex = y to the inverse natural log
equation.
3. 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?
http://pulse.pharmacy.arizona.edu/10th_grade/disease_epidemics/science/medic.html
Teacher’s Guide – The e Handout
4. Solve for “y”
ex = 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.
5. Convert the ex = y to the inverse natural log
equation.
loge(y) = x
6. 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.
http://pulse.pharmacy.arizona.edu/10th_grade/disease_epidemics/science/medic.html