Pre Algebra Change of Log Base
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We're Way Off Base
Change of Log Base

 
 What if we really wanted to find out what that exponent was 
 in the last example on the last page? 
 Remember it?
 

7log7500 = 500

 
 So how would we find the answer?
 
 We could start the way we did on the log5125 problem in the last page:
 

 
 OOPS! Three wasn't enough, and 4 was too many. 
 343 is a lot closer to 500 than 2,401.
 That means the answer is closer to 3 than to 4. 
 We know it's:
 

3.(something)

 
 But what?
 
 We probably can't find a calculator with a base 7 log key. 
 We have to find a way to convert base 7 logs to base 10 logs
 so we can use a calculator or we're stuck.
 
 You're in luck! We just happen to have a formula to do that.
 
 (Somehow I just knew we would!)
 
 So if we have something like:
 

log7500

 
 We want to change this into a base 10 log. 
 To do this, we make a fraction. 
 The fraction has a base 10 log on the top (numerator). 
 
 We put another base 10 log on the bottom (the denominator). 
 We have:
 

 
 That's all there is to it!
 Now if we have a calculator with a log key, we can solve this puppy.
 

 log10500 = 2.69897                log107 = 0.845098

 
 So ...
 

 

 
 So that means ...
 

 73.19368 = 500

 
 In general, using a and b to stand for any old numbers we might have, 
 we can say:
 

 
 This trick works for any base you want to change to. 
 You could even say:
 

 
 if you wanted to. 
 But why would you want to?
 

   copyright 2005 Bruce Kirkpatrick

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