Are Properties of Logarithms Missing Something?

Date: 11/17/2005 at 09:05:28
From: Sara
Subject: logarithms

I have a question about using the properties of logs to solve 
equations.  For example, I can solve this equation in two ways:

  ln x^2 = -7
  e^(ln x^2) = e^(-7)
  x^2 = e^(-7)
  x = plus or minus (e^(-7/2))

or

  ln x^2 = -7
  2 ln x = -7
  ln x = -7/2
  x = e^(-7/2)

Using the second method, you only get the positive answer.  Where did 
I lose the negative answer?  Why is this not taken into account when 
using properties like this are discussed?

Date: 11/17/2005 at 09:45:09
From: Doctor Peterson
Subject: Re: logarithms

Hi, Sara.

Good question!  This is a fact that is sometimes swept under the rug:
when you apply some of these properties, the domain of the expression
changes.  In particular, although

  log(a^n) = n log(a)

for any a and n for which both sides are defined, the left side is 
defined when a < 0 and n is even, but the right side is not.  So in 
applying the property, you lose negative values of a.

Sometimes books mention this, but then avoid it by specifying that the 
variable is positive when giving equations to solve.  Some books may 
not mention it at all, which is not a good idea.

Another way it could be dealt with would be to use the following when 
n is even:

  log(a^n) = n log|a|

Authors probably don't want to subject their students to a mix of 
absolute values and logs, which would be too much for many of us!  But 
this would retain all solutions in your problem:

  ln x^2 = -7
  2 ln |x| = -7
  ln |x| = -7/2
  |x| = e^(-7/2)
  x +- e^(-7/2)

See this page:

  Error in One of the Laws of Logarithms?
    http://mathforum.org/library/drmath/view/60598.html 

If you have any further questions, feel free to write back.


- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/