L4.2 A2 Laws of Logarithms - Part 1
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Unit 4
Logarithms
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Part 4.2A corresponds to section 8.3, starting on page 392 of your Pre-Calculus 12 textbook.
Look at the keys on your calculator.
If you are using a Texas Instruments calculator, you will find the logarithmic keys on the left side.
One key is labelled LOG, and its «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mn»2«/mn»«mi»nd«/mi»«/msup»«/mstyle»«/math» function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mn»10«/mn»«mi»x«/mi»«/msup»«/mstyle»«/math».
The two calculator functions are related because «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mi»log«/mi»«mi»y«/mi»«/mrow»«/mstyle»«/math» expressed in exponential form is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msup»«mn»10«/mn»«mi»x«/mi»«/msup»«/mrow»«/mstyle»«/math».
The key labelled LN is a special logarithm called the natural logarithm.
The natural logarithm is not explored in this course, but if you continue your mathematical studies in calculus, you will learn more about the natural logarithm.
So, for the purposes of this course, the key on your calculator that will be used regularly is the LOG key. It is the logarithm with base «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math».
What happens if the logarithmic expression, function, or equation is not base «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math»? Can the calculator still be used?
It can indeed. Later is the Lesson, we will explore how to convert logarithms of any base to logarithms of base «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math».
If you are using a Texas Instruments calculator, you will find the logarithmic keys on the left side.
One key is labelled LOG, and its «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mn»2«/mn»«mi»nd«/mi»«/msup»«/mstyle»«/math» function is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msup»«mn»10«/mn»«mi»x«/mi»«/msup»«/mstyle»«/math».
The two calculator functions are related because «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»=«/mo»«mi»log«/mi»«mi»y«/mi»«/mrow»«/mstyle»«/math» expressed in exponential form is «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msup»«mn»10«/mn»«mi»x«/mi»«/msup»«/mrow»«/mstyle»«/math».
The key labelled LN is a special logarithm called the natural logarithm.
The natural logarithm is not explored in this course, but if you continue your mathematical studies in calculus, you will learn more about the natural logarithm.
So, for the purposes of this course, the key on your calculator that will be used regularly is the LOG key. It is the logarithm with base «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math».
What happens if the logarithmic expression, function, or equation is not base «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math»? Can the calculator still be used?
It can indeed. Later is the Lesson, we will explore how to convert logarithms of any base to logarithms of base «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»10«/mn»«/mstyle»«/math».
The logarithmic laws are related to the exponent laws, as you discovered in the Warm Up. The table below lists three of the most commonly used laws of logarithms, along with their names.
| Logarithmic Laws | Related Exponent Laws |
|
Product Law of Logarithms
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mi»c«/mi»«/msub»«mi»M«/mi»«mi»N«/mi»«mo»=«/mo»«msub»«mi»log«/mi»«mi»c«/mi»«/msub»«mi»M«/mi»«mo»+«/mo»«msub»«mi»log«/mi»«mi»c«/mi»«/msub»«mi»N«/mi»«/mrow»«/mstyle»«/math» |
Product Law of Exponents
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«msup»«mi»c«/mi»«mi»P«/mi»«/msup»«/mfenced»«mfenced»«msup»«mi»c«/mi»«mi»Q«/mi»«/msup»«/mfenced»«mo»=«/mo»«msup»«mi»c«/mi»«mrow»«mi»P«/mi»«mo»+«/mo»«mi»Q«/mi»«/mrow»«/msup»«/mrow»«/mstyle»«/math» |
|
Quotient Law of Logarithms
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mi»c«/mi»«/msub»«mfenced»«mfrac»«mi»M«/mi»«mi»N«/mi»«/mfrac»«/mfenced»«mo»=«/mo»«msub»«mi»log«/mi»«mi»c«/mi»«/msub»«mi»M«/mi»«mo»§#8722;«/mo»«msub»«mi»log«/mi»«mi»c«/mi»«/msub»«mi»N«/mi»«/mrow»«/mstyle»«/math» |
Quotient Law of Exponents
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mfenced»«mfrac»«msup»«mi»c«/mi»«mi»P«/mi»«/msup»«msup»«mi»c«/mi»«mi»Q«/mi»«/msup»«/mfrac»«/mfenced»«mo»=«/mo»«msup»«mi»c«/mi»«mrow»«mi»P«/mi»«mo»§#8722;«/mo»«mi»Q«/mi»«/mrow»«/msup»«/mrow»«/mstyle»«/math» |
|
Power Law of Logarithms
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mi»c«/mi»«/msub»«msup»«mi»M«/mi»«mi»N«/mi»«/msup»«mo»=«/mo»«mi»N«/mi»«msub»«mi»log«/mi»«mi»c«/mi»«/msub»«mi»M«/mi»«/mrow»«/mstyle»«/math» |
Power Raised to a Power
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msup»«mfenced»«msup»«mi»c«/mi»«mi»P«/mi»«/msup»«/mfenced»«mi»Q«/mi»«/msup»«mo»=«/mo»«msup»«mi»c«/mi»«mrow»«mi»P«/mi»«mo»§#8226;«/mo»«mi»Q«/mi»«/mrow»«/msup»«/mrow»«/mstyle»«/math» |
Note that just as with the laws of exponents, the laws of logarithms apply when we are working with logarithms of like bases.
The laws of logarithms can be used to simplify expressions such that they can be more easily evaluated.
The laws of logarithms can be used to simplify expressions such that they can be more easily evaluated.