Learning Objectives
Introduction
Tutorial
Properties of Logarithms
m > 0 and n > 0
Hmmmm, why don’t I just take the product of their logs??????
Wait a minute, I remember my teacher saying above that logs are another way to write exponents - WHENEVER I WAS MULTIPLYING LIKE BASES, I ADDED MY EXPONENTS - SO I’M GOING TO HAVE TO ADD MY LOGS - EUREKA!!!!
Note that even though m and n are not the bases of the log itself, they can each be written as base b to an exponent, because of the definition of logarithms.
m > 0 and n > 0
So here, we have to remember that when we were dividing like bases, we subtracted our exponents - so we do the same type of thing with our logs.
m > 0
Wow, that looks a little different, but again it comes from the fact that logs are another way to write exponents.
Remember that when we had a base raised to 2 powers that we would multiply those 2 exponents together. That is what we are doing here. Again, even though m is not the base of the log, it can be written as b to an exponent (based on the log definition) and the log itself is an exponent so we have a double exponent - so we multiply our exponents together.
m > 0 and b > 0
Your calculator is limited to only finding base 10 and base e logarithms. That would leave us in a bind if we needed to find the value of a log with any other base. So we can use this change-of-base formula to change it to base 10 or e so we could find a value. Neat, huh?
What would the base be in this problem? If you said 10 you are correct. This is known as the common log.
If you need a review on the common log (log base 10), feel free to go to Tutorial 43: Logarithmic Functions.
What would the base be in this problem? If you said e you are correct. This is known as the natural log.
If you need a review on the natural log (log base e), feel free to go to Tutorial 43: Logarithmic Functions.
This time the base is 5. Make sure that you keep that same base throughout the problem.
*Use the quotient rule
*Use the power
rule
*Use the definition
of logs to simplify
*3 is the exponent needed on 5 to
get 125
This time the base is 2. Make sure that you keep that same base throughout the problem.
*Use the product rule and the quotient rule
*Use the definition
of logs to simplify
*5 is the exponent needed on 2 to
get 32
*Use the power
rule
What is the base in this problem?
This time the base is e. Make sure that you keep that same base throughout the problem.
What is the base in this problem?
This time the base is 3. Make sure that you keep that same base throughout the problem.
*Use the definition
of logs to simplify
*3 is the exponent needed on 3 to
get 27
What is the base in this problem?
This time the base is e. Make sure that you keep that same base throughout the problem.
*Use the product
rule
This should land between 1 and 2, because 7.25 lands between 4^1 = 4 and 4^2 = 16. Again logs are another way to write exponents and that is what we are looking for here.
*Use the calculator to find ln
7.25 and ln
4
*Divide
Practice Problems
To get the most out of these, you should work the problem out on your own and then check your answer by clicking on the link for the answer/discussion for that problem. At the link you will find the answer as well as any steps that went into finding that answer.
Practice Problems 1a - 1c: Expand each logarithmic expression as much as possible. Evaluate without a calculator where possible.
Practice Problems 2a - 2b: Condense each logarithmic expression into one logarithmic expression. Evaluate without a calculator where possible.
Practice Problem 3a: Rewrite the logarithmic expression using natural logarithms and evaluate using a calculator. Round to 4 decimal places.
Need Extra Help on these Topics?
The following are webpages that can assist you in the topics that were covered on this page.
http://www.purplemath.com/modules/logrules.htm
This webpage helps you with logarithmic properties.
http://www.sosmath.com/algebra/logs/log4/log43/log43.html
This webpage helps explain the change of base formula.
Go to Get
Help Outside the
Classroom found in Tutorial 1: How to Succeed in a Math Class for
some
more suggestions.
Last revised on March 23, 2011 by Kim Seward.
All contents copyright (C) 2002 - 2011, WTAMU and Kim Seward. All rights reserved.