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/ |
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