L4.2 A2 Laws of Logarithms - Part 5
Completion requirements
Unit 4
Logarithms
Changing Bases of Logarithms
It is possible to use the formula «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mi»b«/mi»«/msub»«mi»c«/mi»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»log«/mi»«mi»a«/mi»«/msub»«mi»c«/mi»«/mrow»«mrow»«msub»«mi»log«/mi»«mi»a«/mi»«/msub»«mi»b«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math» to change the base of a logarithm from «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»b«/mi»«/mstyle»«/math» to «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mi»a«/mi»«/mstyle»«/math».Explore Your Understanding Assignment 4.2 includes a question that explores the derivation of this formula. This formula is helpful if you want a particular base, such as when you want to simplify a logarithmic expression or you want to enter it into a calculator.
Express «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mn»5«/mn»«/msub»«mn»256«/mn»«/mrow»«/mstyle»«/math» using base «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mn»2«/mn»«/mstyle»«/math» logarithms.
To use «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«msub»«mi»log«/mi»«mi»b«/mi»«/msub»«mi»c«/mi»«mo»=«/mo»«mfrac»«mrow»«msub»«mi»log«/mi»«mi»a«/mi»«/msub»«mi»c«/mi»«/mrow»«mrow»«msub»«mi»log«/mi»«mi»a«/mi»«/msub»«mi»b«/mi»«/mrow»«/mfrac»«/mrow»«/mstyle»«/math», notice that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»b«/mi»«mo»=«/mo»«mn»5«/mn»«/mrow»«/mstyle»«/math», «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»c«/mi»«mo»=«/mo»«mn»256«/mn»«/mrow»«/mstyle»«/math», and «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«mrow»«mi»a«/mi»«mo»=«/mo»«mn»2«/mn»«/mrow»«/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»«msub»«mi»log«/mi»«mn»5«/mn»«/msub»«mn»256«/mn»«/mtd»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mrow»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mn»256«/mn»«/mrow»«mrow»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mn»5«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«mtr»«mtd/»«mtd»«mo»=«/mo»«/mtd»«mtd»«mfrac»«mn»8«/mn»«mrow»«msub»«mi»log«/mi»«mn»2«/mn»«/msub»«mn»5«/mn»«/mrow»«/mfrac»«/mtd»«/mtr»«/mtable»«/mstyle»«/math»
Note that «math style=¨font-family:Verdana¨ xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mstyle mathsize=¨14px¨»«msub»«mi»log«/mi»«mn»5«/mn»«/msub»«mn»256«/mn»«mo»=«/mo»«mn»8«/mn»«/mstyle»«/math». Changing the base doesn’t always result in an expression that can easily be evaluated.