



The
rules of logs look a bit like the rules of exponents.












and ...
(watch closely) 


Log
a^{b} = b × Log a






This is
the most valuable rule about logs. 


It lets
you do an amazing thing ... 





Say you
had the equation: 


200 = 2^{X}






What
exponent do you raise 2 to, to get 200? 


Or to say
it another way ... 


How many
2's do you have to multiply together to get 200? 


(2x2x2x2x
...) 





Here's
the way to solve this one: 





We can do
almost anything we want to do to an equation 


as long
as we do the same thing to both sides of the equation. 


Right
now, we're going to take the log of both sides. 


We can
use any log base we like, 


so chose
a base that we have a key for on our calculator. 


That
probably means using 10 or e. 





200 = 2^{X}



Ln 200 =
Ln 2^{X}






Here it
is. The reason logs are still taught in math. 


When we
have something like the right side of this equation 


we can
change it like this: 





Ln 2^{X}
= X Ln2






That's
right. If you have the log of a term with an exponent, 


you can
move the exponent to the left of the ln or log. 


That
turns the exponent into a plain number factor. 


Ans a
number factor is WAY easier to deal with than an exponent! 








Now back
to the problem ... 





200 = 2^{X}



Ln 200 =
Ln 2^{X}



Ln
200 = X Ln 2






Now
divide both sides by Ln 2 and cancel ... 








Ln 2 and
Ln 200 are just numbers! 


We can
punch them up on a calculator 


and solve
for X ... 








copyright 2005 Bruce Kirkpatrick 
