By applying the rules for natural logarithms, how could 2ln|x| be expressed so that the absolute value symbol is not required? Help?!

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Answer 1 · 2017-08-28T21:46:07

#ln(x^2)#

Apply the power rule for logarithms

#2ln|x|=ln(|x|^2)#

As we know, the square of a positive or negative number is always positive, so

#|x|^2=x^2#

and the answer becomes #ln(x^2)#.

Answer 2 · 2017-08-28T21:50:54

#2ln|x| = ln(x^2); x!=0#

A property of logarithms is:

#cln(a) = ln(a^c)#

If we do this to the given logarithm

#2ln|x| = ln|x^2|#

We can drop the absolute value symbol:

#2ln|x| = ln(x^2)#

Please understand that the absolute value symbol is there

#2ln|x| = ln(x^2)#

to prevent a negative number from being entered into the logarithm. No logarithm can take a negative number. Try this on your calculator:

#ln(-1)#

or any negative number.

You receive an error message.

We can drop the absolute value, because #x^2# can never be negative.

There is still something missing, in either case, #x !=0#

No logarithm can take 0 as an argument. Try it.

#ln(0) = **ERROR** #

Therefore, we must always stipulate that #x != 0# with or without the absolute value symbol.

#2ln|x| = ln(x^2); x!=0#