Pre Algebra Logarithms
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Inside Out Exponents
Logarithms

 
 You could ask the question: 
 "What exponent do you have to raise 5 to, to get 125?"
 and most everybody would know what you meant and be perfectly happy. 
 
 Everyone that is, EXCEPT math types.
 
 Math people wanted a way to write stuff like that in a few numbers and symbols.
 They got one. What they came up with was the word "log." 
 A kind of funny word for math eh? 
 Well who knows why they chose it, but what it means is ...
 "The exponent that you have to raise" 
 after it you write a small number and then a regular size number. 
 The small number is the number you multiply times itself. 
 The big number is the answer you get when you do the multiplying.
 It goes together like this:
 

 

 SO WHAT IS THE EXPONENT 5 MUST BE RAISED TO TO GET 125?

 

 
 So ...
 

 log5125 = 3

 
 The way you SAY this thing in English goes like this:
 
 Fancy way: The log, base 5 of 125 equals 3.
 
 Less fancy way: log 5 of 125 is 3.

 

 But it means: The exponent that 5 must be raised to, to get 125 is 3.
 

 
 Always, always, always read the meaning when you look at logs.
 
 The little number after the word log is called the base. 
 It is very important. 
 
 In ancient times (before 1980) the only easy way 
 to find the value of something like:
 

 log4256

 
 was to try to look it up in a table of logs in a book. 
 
 These tables are lists of logs that were printed 
 in the back of many math books.
 
 Usually, the tables were only printed for two different bases. 
 Base 10 and another special base called e. 
 We'll talk about the e thing some other time.
 
 Base 10 logs are used a bunch. 
 They are used so much that when we write them, 
 we can leave the base number out.
 Like this ...
 

 log10200 = log 200

  
 So if you see:
 

 log 1000

 
 It means:
 

 log101000

 
 And what does log101000 mean?
 
 ALL TOGETHER NOW:
 

 THE EXPONENT THAT YOU HAVE TO RAISE 10 TO,

  TO GET 1000

 

 
 Example:
 
 OK, What if we have ...
 

 log55 = ?

 
 What's the answer?
 Just read the meaning, 
 it means:
 

 THE EXPONENT THAT YOU HAVE TO RAISE 5 TO, 

 TO GET 5

 
 Well:
 

 51 = 5   so   log55 = 1

 

 
 Example:
 
 How about ...
 

 log31 = ?

 
 It means ...
 

 THE EXPONENT THAT YOU HAVE TO RAISE 3 TO, 

 TO GET 1

 
 The special rule says:
 

 30 = 1

 
 So ...
 

 log31 = 0

 

 
 Example:
 
 OK, Try this ...
 

 log12-10 = ?

 
 It means ...
 
 THE EXPONENT THAT YOU HAVE TO RAISE 12 TO, 
 TO GET -10.
 
 Well you can multiply 12's together till the end of time 
 and it will NEVER get a negative number for an answer. 
 
 So this one has no answer.
 
 
 Example:
 
 Here's a very nasty one ...
 

 7log7500 = ?

 
 What does this equal?
 OK, don't panic, just read it. 
 You might get a surprise:
 
 Seven, raised to 
 THE EXPONENT THAT YOU HAVE TO RAISE 7 TO, TO GET 500.
 
 So what do you get 
 when you raise 7 to the exponent that you have to raise 7 to, to get 500?
 

 YOU GET 500!

 

 7log7500 = 500

 
But don't you need to find out what the exponent number is? 
 
 NOPE!
 
 Hey, if they don't ask for it, why do it?
 

   copyright 2005 Bruce Kirkpatrick

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