



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
... 





log_{5}125 =
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: 





log_{4}256






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 ... 





log_{10}200 =
log_{ }200






So if you see: 





log_{
}1000






It means: 





log_{10}1000






And what does log_{10}1000 mean? 





ALL TOGETHER
NOW: 





THE EXPONENT THAT YOU HAVE
TO RAISE 10 TO,



TO GET 1000












Example: 





OK,
What if we have ... 





log_{5}5 =
?






What's the answer? 


Just read the meaning, 


it means: 





THE EXPONENT THAT YOU HAVE
TO RAISE 5 TO,



TO GET 5






Well: 





5^{1} = 5
so log_{5}5 = 1












Example: 





How
about ... 





log_{3}1 =
?






It
means ... 





THE EXPONENT THAT
YOU HAVE TO RAISE 3 TO,



TO GET
1






The special rule says: 





3^{0} =
1






So
... 





log_{3}1 =
0












Example: 





OK,
Try this ... 





log_{12}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 ... 





7^{log}^{7}^{500}
= ?






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!






7^{log}^{7}^{500}
= 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 
