Question

what's the difference between y=lnx and y=abs(lnx)?

Answer 1

type it in your calculator and instead of x type in any number for instance 2 and see what you get in both cases

Answer 2

i typed both of them on the graph. calc and one of them the tip is poiting up and the other points down

Answer 3

logarithms only have a domain of\[(0,\infty)\]so by defining the absolute value we can change the domain to \((-\infty,\infty)\)

Answer 4

the shape of the graph of the absolute value version will be symmetrical about the y-axis

Answer 5

I believe that only applies by changing x for abs(x).
The domain of ln(x) would still be the same even with the absolute value.
abs(ln(x)) at x=-1 => abs(ln(-1)) ?

Answer 6

With these absolute values where they are, the range of ln(x) would be affected, since ln(x) is no longer allowed to return a negative value like when 0 < x <1.

Answer 7

http://www.wolframalpha.com/input/?i=graph+lnx%2C+abs%28lnx%29

Answer 8

\[\ln x=y\implies x=e^y\]now ask yourself, "what value of y would give us a negative value of x?"

if we make y large and negative, x approaches zero, but never a negative number

hence for\[f(x)=\ln x\]x cannot be negative, and the domain is \((0,\infty)\)

Answer 9

Yes, that is true, I just don't think your statement above that the domain changes with the addition of these absolute values to be correct:
\( \color{red}\lvert \ln x \color{red}\rvert = y\)

I think you are referring to this situation, where the domain change because the absolute value of x is evaluated first:
\( \ln \color{red} \lvert x \color{red}\rvert = y\)
but the question specifies the former case.

Answer 10

is the domain for both of them the same then?

Answer 11

Sorry, Turing; medal goes to Access. Because he's not 99 yet. Haha, socialism.

Answer 12

Yes, the domains of each function are the same because \(\ln x\) has the same restrictions of x-values even when placed into the absolute values. You may show this in a similar way to \(\ln x\) shown above if you said \(y = \lvert \ln x \rvert \implies \pm y = \ln x\) and continued normally.


The graph's behavior that you noted ("one of them the tip is poiting up and the other points down") is relevant to how the absolute value affects \(y = \ln x\).
Notice how this behavior only occurs where \(y = \ln x\) is negative (the part of the domain, 0 < x < 1), but the value of \(y = \lvert \ln x \rvert\) there is still positive. The absolute value essentially makes the function's y-value always-positive.

Answer 13

actually Access was completely right and I was completely wrong, so the medals should all go to him :P