# PROPERTIES OF LOGARITHMS

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```					                      PROPERTIES OF LOGARITHMS

1.   log a 1  0

2.   loga a  1
loga M
3.   a            M

4.   loga a r  r

5.   loga ( MN )  loga M  loga N

F I  log M  log
M
6.   log
HK
a
N
a           aN

F I   log N
1
7.   log
HK
N
a                    a

8.   loga M  r loga M
r
9.
Definition:

y  loga x                                   if and only if                             x  ay
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10.

The inverse to an exponential function                               ya             x
is logarithmic function with base    .                    a
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11.

If          a u  av,                     then              u  v.
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12.

Theorem:

MN                       if & only if               loga M  loga N
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13.

Change of base formula:

log b M
loga M 
log b a
Introduction

“This Webquest is designed to bring the logarithms out of the dark and into the light. At this point, logarithms
may just be a name for another set of steps (algorithms) Mrs. Edwards wants you to follow. Upon completion
you will know the importance of logarithms, the history behind them and some places were logarithms appear.”

*What is the definition of a logarithm in terms of an exponential expression?

*What is the name of the inverse function to exponential function?

Hint: Look at the keys on your calculator. Notice the relationship between what is on the key and what is above it.

*How do you know the inverse function you listed above is really the inverse of an exponential function?
Discuss the way their graphs look and domain/range to answer the question.

“What you may not know is that the relationship with exponents wasn’t even realized until years after the
invention of logarithms. Indeed, there was another reason that led mathematicians to develop the idea of a
logarithm, and part of your task is to identify those mathematicians and the reasons behind their work.

There are many contexts in which logarithms are used today. Most commonly, they are used to design certain
measurement scales. Other uses of logarithms that were very common in the past have now been programmed
into machines, so that we do not actually see the logarithms on the surface. In this day and age, one encounters
many logarithms without even realizing it, because they do most of their work behind the scenes. The other

Written Report                                                                           Groups 3-5 Students
Each group will turn in one report. For your research, you may use the links to the websites that I have
provided as well as any books or other outside sources. A list of references must be included at the end
of your report along with the names of the students who are in your group. Attach your report to an e-
mail. Subject line: Class period followed by names of those students in your group.

1. First research the history of the logarithm. Write 3-4 pages, double spaced, 12 font, 1 inch margins,
a. Who were the mathematicians responsible for inventing logarithms?
b. What reason led them to invent logarithms?
c. Who was (were) the mathematician(s) responsible for adapting logarithms into the system
that we recognize today?
d. Write about any other historical topic relating to logarithms that you find interesting. (The
discovery and uses of the natural base “e” is one such possibility.) Only two groups per
topic.
2. Find three applications of logarithms in real-life situations that you find interesting. Include the
following:
a. Describe the application.
b. Your reason for choosing the application.
c. Explain exactly where the logarithm appears in the application and how it works
mathematically. You will need to use all the knowledge you have about logarithmic
functions and possibly exponential functions here. This section must include a mathematical
equation describing the process, with all variables clearly identified. Though not required,
you can discuss any other aspects of the situation that you find interesting. For example,
explain why the use of a logarithm is convenient in your particular context or if the use of
logarithms could lead to misconceptions by those who are not aware of its role.
Oral Presentation
1. Introduction: Use info from step 1 in your report.
2. Thorough description of one of the applications you chose in step 2 of your report.
3. Outline of your presentation. (e-mail me this portion)
Problem Set (E-mail me problem set. / Solutions may be written by hand.)
1. Your group will create the problems and answers to a homework assignment. The problem set should
include one problem that represents each of the log properties in addition to one challenge problem that
requires several properties to solve. The problem set will be given to your classmates the same day you
present.
2. The day after you present.
a. Collect the assignment
3. Two days following your presentation.
a. Hand back graded assignments with solutions that include step by step work leading to the final
b. Announce what percent of the class was able to correctly answer your challenge problem and
explain the steps necessary to complete the problem. Include in your explanation the properties
needed to complete each step. (Group with the lowest percent will earn 5% bonus on their group
c. Record homework scores on grade sheet and hand in to Mrs. Edwards.
Test
*50% the homework assignment that your group handed out to classmates.
*50% problems Mrs. Edwards’ selects from all groups’ problem sets.
*Challenge problems will show up as chances to earn test bonus points.
(Note to group members of 5% bonus on presentation: If you do not get your own group’s challenge
problem correct on the test then you will lose the 5% bonus on your presentation.)

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