# Thread: Does (ln x)^2 = ln x^2?

1. ## Does (ln x)^2 = ln x^2?

If it doesn't, what does it equal?

2. Originally Posted by Truthbetold
If it doesn't, what does it equal?
No.

3. Lol! Equals 'no'. I'll take that as "simplifying it more will be overly difficult if not impossible, so don't bother."

4. Originally Posted by Truthbetold
Lol! Equals 'no'. I'll take that as "simplifying it more will be overly difficult if not impossible, so don't bother."
all that's needed here is a counter example, try x = 2. you will realize you get different answers for each expression

$(\ln x)^2 = \ln x \cdot \ln x$

while $\ln x^2 = 2 \ln |x|$ ....clearly these are not the same thing

5. ln x^2 = 2lnx
(lnx)^2 = (ln x)*(lnx)
so no they are not equal (there may be a value for which they are but that would be a fluke

6. Originally Posted by suissa
ln x^2 = 2lnx
(lnx)^2 = (ln x)*(lnx)
so no they are not equal (there may be a value for which they are but that would be a fluke
$x=1$ or $x=e^2$, no fluke.

7. Originally Posted by DivideBy0
$x=1$ or $x=e^2$, no fluke.
The OP was asking if they were equal, which assumes for all values of x, but it was a good idea to point that out for him/her.

8. Originally Posted by Truthbetold
If it doesn't, what does it equal?
I would view this as an exercise in order of operations:
$lnx^2 = ln(x^2)$
by definition, and since ln is not a linear operator
$(ln(x))^2 \neq ln(x^2)$

-Dan

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# ln(x-2)

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