Indices & logarithm

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  • 1. Introduction ToLogarithms

2. Logarithms were originally developed to simplify complexarithmetic calculations.They were designed to transformmultiplicative processesinto additive ones. 3. If at first this seems like no big deal,then try multiplying 2,234,459,912 and 3,456,234,459.Without a calculator ! Clearly, it is a lot easier to addthese two numbers. 4. Today of course we have calculatorsand scientific notation to deal with such large numbers. So at first glance, it would seem thatlogarithms have become obsolete. 5. Indeed, they would be obsolete except for onevery important property of logarithms. It is called the power property and we will learn about it in another lesson.For now we need only to observe that it is an extremely important partof solving exponential equations. 6. Our first job is totry to make somesense oflogarithms. 7. Our first question thenmust be:What is a logarithm ? 8. Of course logarithms havea precise mathematicaldefinition just like all terms inmathematics. So lets start with that. 9. Definition of Logarithm Suppose b>0 and b1,there is a number p such that:plogb n p if and only if b n 10. Now a mathematician understands exactlywhat that means.But, many a student is leftscratching their head. 11. The first, and perhaps the most important step, inunderstanding logarithms is to realize that they always relate back to exponentialequations. 12. You must be able to convertan exponential equation into logarithmic form and viceversa.So lets get a lot of practice with this ! 13. Example 1:3Write 2 8 in logarithmic form.Solution: log2 8 3We read this as: the logbase 2 of 8 is equal to 3. 14. Example 1a:2Write 4 16 in log arithmic form.Solution:log4 162Read as: the log base 4 of 16 isequal to 2. 15. Example 1b: 3 1 Write 2 in log arithmic form. 81Solution:log2 38 1Read as: "the log base 2 of is equal to -3". 8 16. Okay, so now its time foryou to try some on yourown. 21. Write 7 49 in log arithmic form.Solution: log 7 49 2 17. 02. Write 51 in log arithmic form.Solution:log5 10 18. 213. Write 10 in log arithmic form.100 1Solution: log10 2100 19. 1 4. Finally, write 1624in log arithmic form.1Solution:log16 42 20. It is also very important to beable to start with a logarithmic expression and change this into exponential form. This is simply the reverse of what we just did. 21. Example 1:Write log3 81 4 in exp onential form4 Solution: 3 81 22. Example 2: 1Write log2 3 in exp onential form. 8 3 1Solution:2 8 23. Okay, now you try these nextthree.1. Write log10 1002 in exp onential form. 12. Write log53 in exp onential form.125 13. Write log27 3 in exp onential form. 3 24. 1. Write log10 100 2 in exp onential form.2 Solution:10100 25. 12. Write log5 3 in exp onential form.125 31Solution:5 125 26. 13. Write log27 3 in exp onential form. 3 1Solution: 27 3 3 27. We now know that a logarithm is perhaps best understood as being closely related to anexponential equation.In fact, whenever we get stuckin the problems that follow we will return to this one simple insight.We might even state asimple rule. 28. When working with logarithms, if ever you get stuck, try rewriting the problem inexponential form. Conversely, when workingwith exponential expressions,if ever you get stuck, try rewriting the problemin logarithmic form. 29. Lets see if this simple rulecan help us solve some of the followingproblems. 30. Example 1Solve for x: log 6 x2 Solution:Lets rewrite the problemin exponential form. 26x Were finished ! 31. Example 21Solve for y: log5 y25Solution: Rewrite the problem inexponential form.y 115Since 255 225 5y 52y 2 32. Example 3Evaluate log3 27.Solution:Try setting this up like this: log3 27y Now rewrite in exponential form.y3 27y33 3y 3 33. These next two problems tend to be some of thetrickiest to evaluate.Actually, they are merely identities and the use of our simplerule will show this. 34. Example 4 2 Evaluate: log7 7Solution:log7 72 y First, we write the problem with a variable.y 27 7Now take it out of the logarithmic form and write it in exponential form. y2 35. Example 5log 4 16 Evaluate: 4Solution: log 4 16 4y First, we write the problem with a variable. log4 y log4 16Now take it out of the exponential formand write it in logarithmic form. Just like 23 8 converts to log2 8 3 y 16 36. Ask your teacherabout the last two examples.They may showyou a nice shortcut. 37. Finally, we want to take a look at the Property of Equality for Logarithmic Functions.Suppose b0 and b 1.Then logb x1 log b x 2 if and only if x1x2Basically, with logarithmic functions,if the bases match on both sides of the equalsign , then simply set the arguments equal. 38. Example 1 Solve: log3 (4x 10)log3 (x 1)Solution:Since the bases are both 3 we simply setthe arguments equal. 4x 10 x 1 3x 10 13x9 x3 39. Example 2 2Solve:log8 (x 14) log8 (5x)Solution:Since the bases are both 8 we simply set the arguments equal. 2x 14 5x2x 5x 14 0Factor(x 7)(x 2) 0(x 7) 0 or (x 2) 0 x 7 or x 2 continued on the next page 40. Example 2continued 2 Solve:log8 (x 14)log8 (5x)Solution:x7 or x2 It appears that we have 2 solutions here. If we take a closer look at the definition of a logarithm however, we will see that not only must we use positive bases, but also we see that the arguments must be positive as well. Therefore -2 is not a solution. Lets end this lesson by taking a closer look at this. 41. Our final concern then is todetermine why logarithms likethe one below are undefined. log 2 ( 8)Can anyone give us an explanation ? 42. log 2 ( 8) undefinedWHY? One easy explanation is to simply rewritethis logarithm in exponential form. Well then see why a negative value is not permitted.log 2 ( 8)y First, we write the problem with a variable.y 28 Now take it out of the logarithmic formand write it in exponential form.What power of 2 would gives us -8 ?3 3 1 28 and 28Hence expressions of this type are undefined. 43. That concludes our introduction to logarithms. In the lessons tofollow we will learn some important properties of logarithms. One of these properties will give us a very important tool whichwe need to solve exponentialequations. Until then letspractice with the basic themesof this lesson.