MHB Does e^{2 \ln{|x|}} = |x^2| or x^2?

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The equation e^{2 ln{|x|}} simplifies to x^2, as the natural logarithm and exponential functions cancel each other out. The expression |x^2| is equivalent to x^2 since squaring any real number results in a non-negative value. Therefore, both e^{2 ln{|x|}} and |x^2| yield the same result, confirming their equality. This demonstrates the properties of logarithms and exponents in relation to absolute values. Ultimately, e^{2 ln{|x|}} is equal to x^2.
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is [math]e^{2 \ln{|x|}} = |x^2|[/math] or [math]x^2[/math]?
 
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find_the_fun said:
is [math]e^{2 \ln{|x|}} = |x^2|[/math] or [math]x^2[/math]?

Both, since squaring makes everything non-negative anyway...
 

Thread 'Area of Overlapping Squares'

Here is a little puzzle from the book 100 Geometric Games by Pierre Berloquin. The side of a small square is one meter long and the side of a larger square one and a half meters long. One vertex of the large square is at the center of the small square. The side of the large square cuts two sides of the small square into one- third parts and two-thirds parts. What is the area where the squares overlap?
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