Algebra 2 Restatement of Log Equations
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Logging Further On
Restatement of Log Equations

 

 Example:

 What do we do with this one?

 

 log(X + 3) + log(2X - 5) = 2

 
 Remember the rule of logs that said:
 

 log(a b) = log a + log b

 
 We're going to use that rule,
 but going in the opposite direction ...
 

 log a + log b = log(a b)

 
 So:

 log(X + 3) + log(2X - 5) = 2

 log((X + 3)(2X - 5)) = 2

 

 Multiply it out ... 

 log(2X2 + X - 15) = 2

 
 OK, now what?
 
 Well remember that when we write log we really mean log 10
 so let's write the problem like that ...
 

 log10(2X2 + X - 15) = 2

 
 Now what?
 
 When in doubt, read it out!
 Read the meaning of the statement
 

 
 Or, written without logs ...
 

 
 And while we don't have any tricks to help us solve
 something like log10(2X 2 + X - 15) = 2
 we can solve:

 102 = 2X2 + X - 15

 
 Here goes ...

 

 
 OK, where did we leave that old quadratic formula ...
 
 Here it is:

 A = 2   B = 1   C = - 115

 

 
 Now we take our two answers back to the original problem
 and see if they make sense ...
 

 log(X + 3) + log(2X - 5) = 2

 X = 7.34

 log(7.34 + 3) + log(2(7.34) - 5) = 2

 log(10.34) + log(9.68) = 2

 

 Punch these two up on your calculator and you get ...
 

 1.01 + .99 = 2

 
 It checks!
 
 Now try the other one ...
 

 log(X + 3) + log(2X - 5) = 2

 X = -7.34

 log(-7.34 + 3) + log(2(-7.34) - 5) = 2

 log(-4.83) + log(-20.66) = 2

 
 AND HERE WE HAVE A BIG PROBLEM
 
 Do you remember what logs mean?
 

 

 
 And there's no exponent in the world that we can raise 10 to
 to get minus anything.
 The only thing we can get is positive numbers.
 Even negative exponents give us positive numbers.
 That means if you have something like:
 

 log10X

 

 X MUST BE POSITIVE

 
 So X = -7.83 does NOT check. It is not a solution to the problem.
 
 The numbers that X can be are called it's DOMAIN
 So for stuff like Log X (and ln X too), 
 the domain is all real numbers greater than zero.
 
 If we had ...

 Y = ln(X-1)

 
 The (X-1) part all together must be positive.
 That means X - 1 > 0
 
 So:

 

 
 The "domain" of X in Y = ln(X-1) is X >1 "all real numbers greater than 1"
 

   copyright 2005 Bruce Kirkpatrick

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