DISCOVERING ALGEBRA WITH
GRAPHING CALCULATORS
Investigating Logarithmic Functions
Teacher’s Guide
Grade Level: 9–12
Curriculum Focus: Math
Running Time: 25 minutes
Program Description
Examines logarithmic functions and their use in astronomy and other sciences. Students learn
to graph logarithmic functions and their inverses. An archaeologist reveals how logarithms
factor into radiocarbon dating of artifacts.
Learning Objectives
After viewing the program and participating in discussion, students will be able to:
· Explain the concept of a logarithm and relate logarithms to exponents;
· Characterize the graph of a logarithmic function;
· Define an irrational number as a number that cannot be expressed as the quotient of
two integers;
· Consider logarithmic applications in scientific and mathematical contexts;
· Apply algebra to real-world situations and develop logical reasoning skills.
Classroom Connections
When were logarithms first applied to physics and biology? What famous scientists used
logarithms to formulate their theories?
How are logarithms and exponents related? List other examples of inverses in mathematics
and other subjects.
What are the domain and range for logarithmic functions? How are these values related to the
domain and range for exponential functions?
Why is the logarithm of a negative number undefined?
What is the difference between a natural log and a common log? What is an irrational number?
Published by Discovery Education. © 2007. All rights reserved.
How are logarithms used in mathematics and science? Other than the examples given in the
program, where else to logarithms appear in the world?
Classroom Activities
This activity works with exponential and logarithmic functions. Break the class into
pairs, and give the groups the following scenario: you are working in a microbiology lab and
observing the mitosis patterns of paramecia in Petri dishes. In one experiment, you started
with one cell and observed the cells doubling every minute. Working with a partner, find
solutions to these problems.
-Write an equation with base two to determine the number (population) of cells after
one hour.
-Determine the number of cells after one hour.
-Determine how long it would take the population (number of cells) to reach 100,000
cells by graphing the function. (Hint: use the natural logarithm.)
Working in the same groups, students should try this base 10 log problem: In 1974, a
leading scientific magazine said that the rate of energy consumption increased 3.1 percent per
year between 1947 and 1971. If the coal reserves would last 600 years at the 1972 consumption
rate, how long would they last if consumption continued to increase at the rate of 3.1 percent
per year?
Target Vocabulary*
Cartesian coordinate - either of two coordinates that locate a point on a plane and measure its
distance from either of two intersecting straight-line axes along a line parallel to the other axis
function - a: a mathematical correspondence that assigns exactly one element of one set to each
element of the same or another set; b: a variable (as a quality, trait, or measurement) that
depends on and varies with another
inverse - a set element that is related to another element in such a way that the result of
applying a given binary operation to them is an identity element of the set
logarithm - the exponent that indicates the power to which a base number is raised to produce
a given number (the logarithm of 100 to the base 10 is 2)
logarithmic function - a function (as y = loga x or y = ln x) that is the inverse of an exponential
function (as y = ax or y = ex) so that the independent variable appears in a logarithm
quadrant - any of the four parts into which a plane is divided by rectangular coordinate axes
lying in that plane
Page 2
Published by Discovery Education. © 2007. All rights reserved.
y-intercept - the y-coordinate of a point where a line, curve, or surface intersects the y-axis
*All definitions from Merriam-Webster Online: http://www.m-w.com
Academic Standards
The National Council of Teachers of Mathematics (NCTM) has developed national standards
to provide guidelines for teaching mathematics. To view the standards online, go to
http://standards.nctm.org/.
This lesson plan addresses the following math standards:
· Represent and analyze mathematical situations using algebraic symbols
· Understanding patterns, relations, and functions
Page 3
Published by Discovery Education. © 2007. All rights reserved.