Error in One of the Laws of Logarithms?
Date: 05/02/2002 at 13:59:30 From: Charles Pence Subject: Error in one of the laws of logarithms? We were discussing a problem in precalculus today and seemed to discover a basic flaw in one of the exponent laws. Recall: log(x^2) = 2log(x) It is also a fact that the log functions have a domain restriction on all values less than or equal to zero. However: For log(x^2), the only value that is restricted (less than or equal to zero) is zero itself (log(0^2) = log(0)). For 2log(x), which should supposedly be the same function by the law stated above, there are restrictions on all values of X less than or equal to zero. Basically, what it comes down to is that negative numbers are an acceptable input for the function before you apply the law, and negative numbers are no longer an acceptable input after you apply the law. This means that the two functions are not the same, and inherently disproves that law of logarithims. Doesn't it? We're thoroughly perplexed, and resorted to the Internet for assistance. Thanks, Charles Pence
Date: 05/02/2002 at 16:12:49 From: Doctor Peterson Subject: Re: Error in one of the laws of logarithms? Hi, Charles. You haven't shown that the law is wrong, but only that it has an implicit restriction: log(a^b) = b log(a) for all a and b for which both logarithms are defined If a is negative and b even, then the left side is defined but the right side is not. You are taking the property to mean that the function on the left is identical to the function on the right, including having the same domain; but that's not what it means. It is only a pointwise identity (true for one pair of values at a time), not a statement about the two functions as a whole. The same can be said of the other logarithm identities, such as log(ab) = log(a) + log(b) where, if a and b are both negative, the left side is defined but the right is not. We even face the same problem with simpler facts: (sqrt(x))^2 = x is true whenever sqrt(x) is defined; it is not wrong just because there are values of x for which only the right side is defined. We just have to clearly state the restriction: "for all x >= 0" . - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/
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