



What if we really wanted to find out what that exponent
was 


in the last example
on the last page? 


Remember
it? 





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





So how would we find the answer? 





We could start the way we did on the log_{5}125
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: 





log_{7}500






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. 





log_{10}500 =
2.69897
log_{10}7 = 0.845098






So
... 











So
that means ... 





7^{3.19368} =
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 
