L2 Review of Logarithms and Exponential Functions - Part 4
Completion requirements
Unit 6
Exponential and Logarithmic Functions
Lesson 2: Review of Logarithms and Exponential Functions
Expand «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«mfrac»«mrow»«mn»7«/mn»«mi»x«/mi»«/mrow»«mroot»«mi»y«/mi»«mn»4«/mn»«/mroot»«/mfrac»«/mfenced»«/mrow»«/mstyle»«/math».
Three of the logarithmic laws can be used in the expansion: the product law, the quotient law, and the power law.
First, expand by applying the quotient law of logarithms.
Recall «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mroot»«mi»y«/mi»«mn»4«/mn»«/mroot»«mo»=«/mo»«msup»«mi»y«/mi»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/msup»«/mrow»«/mstyle»«/math».
Now, expand using the product law of logarithms.
And, expand further using the power law of logarithms.
First, expand by applying the 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»«mn»3«/mn»«/msub»«mfenced»«mfrac»«mrow»«mn»7«/mn»«mi»x«/mi»«/mrow»«mroot»«mi»y«/mi»«mn»4«/mn»«/mroot»«/mfrac»«/mfenced»«mo»=«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«mrow»«mn»7«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#8722;«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«mroot»«mi»y«/mi»«mn»4«/mn»«/mroot»«/mfenced»«/mrow»«/mstyle»«/math»
Recall «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mroot»«mi»y«/mi»«mn»4«/mn»«/mroot»«mo»=«/mo»«msup»«mi»y«/mi»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/msup»«/mrow»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«mrow»«mn»7«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#8722;«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«mroot»«mi»y«/mi»«mn»4«/mn»«/mroot»«/mfenced»«mo»=«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«mrow»«mn»7«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#8722;«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«msup»«mi»y«/mi»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/msup»«/mfenced»«/mrow»«/mstyle»«/math»
Now, expand using the 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»«mn»3«/mn»«/msub»«mfenced»«mrow»«mn»7«/mn»«mi»x«/mi»«/mrow»«/mfenced»«mo»§#8722;«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«msup»«mi»y«/mi»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/msup»«/mfenced»«mo»=«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»7«/mn»«mo»+«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mi»x«/mi»«mo»§#8722;«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«msup»«mi»y«/mi»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/msup»«/mfenced»«/mrow»«/mstyle»«/math»
And, expand further using the 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»«mn»3«/mn»«/msub»«mn»7«/mn»«mo»+«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mi»x«/mi»«mo»§#8722;«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mfenced»«msup»«mi»y«/mi»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«/msup»«/mfenced»«mo»=«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mn»7«/mn»«mo»+«/mo»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mi»x«/mi»«mo»§#8722;«/mo»«mfrac»«mn»1«/mn»«mn»4«/mn»«/mfrac»«msub»«mi»log«/mi»«mn»3«/mn»«/msub»«mi»y«/mi»«/mrow»«/mstyle»«/math»
The laws of logarithms can also be used to simplify expressions to a single logarithm.
Rewrite «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn»3«/mn»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mn»5«/mn»«mo»§#8722;«/mo»«mn»4«/mn»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mfenced»«mrow»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mi»t«/mi»«/mrow»«/mstyle»«/math»
as a single logarithm.
First, identify where the power law of logarithms can be applied. If there is a coefficient on a «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»l«/mi»«mi»o«/mi»«mi»g«/mi»«/mstyle»«/math»
term, the power law of logarithms can be applied.
Now, using the product and quotient laws, write the expression as a single logarithm.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mn mathcolor=¨#FF0000¨»3«/mn»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mn»5«/mn»«mo»§#8722;«/mo»«mn mathcolor=¨#0080FF¨»4«/mn»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mfenced»«mrow»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»+«/mo»«mfrac
mathcolor=¨#7F007F¨»«mn»1«/mn»«mn»2«/mn»«/mfrac»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mi»t«/mi»«mo»=«/mo»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«msup»«mn»5«/mn»«mn mathcolor=¨#FF0000¨»3«/mn»«/msup»«mo»§#8722;«/mo»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«msup»«mfenced»«mrow»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn
mathcolor=¨#0080FF¨»4«/mn»«/msup»«mo»+«/mo»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«msup»«mi»t«/mi»«mfrac mathcolor=¨#7F007F¨»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mrow»«/mstyle»«/math»
Now, using the product and quotient laws, write the expression as a single logarithm.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mn»3«/mn»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mn»5«/mn»«mo»§#8722;«/mo»«mn»4«/mn»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mfenced»«mrow»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mo»+«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mi»t«/mi»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«msup»«mn»5«/mn»«mn»3«/mn»«/msup»«mo»§#8722;«/mo»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«msup»«mfenced»«mrow»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»4«/mn»«/msup»«mo»+«/mo»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«msup»«mi»t«/mi»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mfenced»«mfrac»«mrow»«msup»«mn»5«/mn»«mn»3«/mn»«/msup»«mo»§#8729;«/mo»«msup»«mi»t«/mi»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»4«/mn»«/msup»«/mfrac»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mi»or«/mi»«/mtd»«mtd/»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mfenced»«mfrac»«mrow»«mn»125«/mn»«msqrt»«mi»t«/mi»«/msqrt»«/mrow»«msup»«mfenced»«mrow»«msup»«mi»t«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mn»1«/mn»«/mrow»«/mfenced»«mn»4«/mn»«/msup»«/mfrac»«/mfenced»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
| Note: If a positive sign is in front of a «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»log«/mi»«/mstyle»«/math» term, that term will appear in the numerator. If a negative sign is in front a «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»log«/mi»«/mstyle»«/math» term, that term will appear in the denominator. |
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