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Say we had some equation $x$ = $y$, we can conclude that $|x| = |y|$, for definite, but if we started with $|x| = |y|$, we cannot go and say $x=y$, as there's the possibility that $x=-y$. This is true, I think. Now let's look at another situation, where we have $\ln|x+5| = \ln|y|$, in my textbook (A Level Maths year 2 pure, UK specification), the textbook concludes that $y=x+5$, but to me this makes no sense because when we had the $|x| = |y|$ situation we couldn't conclude that $x=y$, so why is it any different when we are using the $\ln(x)$ function?

lulu
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Nav Bhatthal
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    Context is critical here. As you say, if we know nothing further about $x,y$ then all you can say is that $(x+5)=\pm y$. But I expect that your textbook included further information. Can you link to it? – lulu Jan 04 '23 at 11:45
  • I'll take a picture when I'm home but this question just sprung to mind. – Nav Bhatthal Jan 04 '23 at 11:55
  • And of course it could be a careless error in the reference, these certainly do happen. But I expect that once you look closely you'll find that more was known about $x$ and $y$. – lulu Jan 04 '23 at 11:57
  • Ah I've found the solution. It's because of the constant of integration which allows for the ambiguity, as the constant has no fixed sign – Nav Bhatthal Jan 04 '23 at 15:31

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