What Does e^ln|x| Equal: x or |x|?

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The expression e^ln|x| equals |x| for any real number x. This is because the natural logarithm function, ln, is only defined for positive values, and thus ln|x| is used to ensure the argument is valid. For instance, when x = -1, ln|x| results in ln(1) = 0, leading to e^ln|x| = e^0 = 1, which is equal to |-1|. Therefore, e^ln|x| consistently yields the absolute value of x.

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If I take e^ln|x|, do I get x or |x|?
 
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|x|

For example, if x = -1, ln(x) is undefined, but ln|x| = 0, and e^ln|x| = e^0 = 1 = |-1|
 
e(any real number) is always positive.
 

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