L4.2 A2 Laws of Logarithms - Part 2
Completion requirements
Unit 4
Logarithms
Evaluate the logarithmic expression «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mn»6«/mn»«mo»+«/mo»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mn»2«/mn»«/mrow»«/mstyle»«/math» by first expressing it as a single logarithm using the product law of logarithms.
Now, the logarithm can be converted to exponential form and evaluated using graphing technology.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»Let«/mi»«mo»§#160;«/mo»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mfenced»«mn»12«/mn»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math».
Use technology to sketch the graphs of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»12«/mn»«/mrow»«/mstyle»«/math», and then determine the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-coordinate of the point of intersection of the two graphs.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mo»§#160;«/mo»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mn»6«/mn»«mo»+«/mo»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mn»2«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mfenced»«mn»6«/mn»«/mfenced»«mfenced»«mn»2«/mn»«/mfenced»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mfenced»«mn»12«/mn»«/mfenced»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Now, the logarithm can be converted to exponential form and evaluated using graphing technology.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«mi»Let«/mi»«mo»§#160;«/mo»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mfenced»«mn»12«/mn»«/mfenced»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mi»x«/mi»«/mtd»«/mtr»«/mtable»«/mstyle»«/math».
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mtable columnalign=¨right center left¨»«mtr»«mtd»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mn»12«/mn»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Use technology to sketch the graphs of «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«msup»«mn»2«/mn»«mi»x«/mi»«/msup»«/mrow»«/mstyle»«/math» and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»y«/mi»«mo»=«/mo»«mn»12«/mn»«/mrow»«/mstyle»«/math», and then determine the «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»x«/mi»«/mstyle»«/math»-coordinate of the point of intersection of the two graphs.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»x«/mi»«mo»§#8784;«/mo»«mn»3«/mn»«mo».«/mo»«mn»58«/mn»«/mrow»«/mstyle»«/math»
Expand «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mfrac»«mrow»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mi»y«/mi»«/mrow»«msqrt»«mi»z«/mi»«/msqrt»«/mfrac»«/mfenced»«/mrow»«/mstyle»«/math».
Three of the logarithmic laws are used, the quotient law, the product law, and the power law.
First, expand by applying the quotient law of logarithms.
Recall that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msqrt»«mi»z«/mi»«/msqrt»«mo»=«/mo»«msup»«mi»z«/mi»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mrow»«/mstyle»«/math».
The expression is now written as follows.
Now, expand by applying the product law.
And, simplify further using the power law.
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»«mo»=«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mi»y«/mi»«/mrow»«/mfenced»«mo»-«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«msqrt»«mi»z«/mi»«/msqrt»«/mfenced»«/mrow»«/mstyle»«/math»
Recall that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msqrt»«mi»z«/mi»«/msqrt»«mo»=«/mo»«msup»«mi»z«/mi»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mrow»«/mstyle»«/math».
The expression is now written as follows.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»=«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mrow»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«mi»y«/mi»«/mrow»«/mfenced»«mo»-«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«msup»«mi»z«/mi»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mfenced»«/mrow»«/mstyle»«/math»
Now, expand by applying the product law.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»=«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«msup»«mi»x«/mi»«mn»5«/mn»«/msup»«/mfenced»«mo»+«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mi»y«/mi»«/mfenced»«mo»-«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«msup»«mi»z«/mi»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«/msup»«/mfenced»«/mrow»«/mstyle»«/math»
And, simplify further using the power law.
«math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mo»=«/mo»«mn»5«/mn»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mi»x«/mi»«/mfenced»«mo»+«/mo»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mi»y«/mi»«/mfenced»«mo»-«/mo»«mfrac»«mn»1«/mn»«mn»2«/mn»«/mfrac»«msub»«mi»log«/mi»«mn»7«/mn»«/msub»«mfenced»«mi»z«/mi»«/mfenced»«/mrow»«/mstyle»«/math»