Properties of natural log and absolute value

[mech-eng]

Homework Statement

from -lnIxI to lnI x^-1I , I try to go from -lnIxI to lnI x^-1I by using some properties.

Homework Equations

- lnIxI

The Attempt at a Solution

First I write the -lnIxI as -1*lnIxI and then I use -1 as an exponent to absolute value of x in the natural log that is
ln ( IxI^-1) and then I take -1 inside the paranthesis and I arrive to ln Ix^-1I which is also ln I 1/x I.

Are my steps correct? If not, which step is incorrect?

Thank you.

 

[Mark44]

Homework Statement

from -lnIxI to lnI x^-1I , I try to go from -lnIxI to lnI x^-1I by using some properties.

Homework Equations

- lnIxI

The Attempt at a Solution

First I write the -lnIxI as -1*lnIxI and then I use -1 as an exponent to absolute value of x in the natural log that is
ln ( IxI^-1) and then I take -1 inside the paranthesis and I arrive to ln Ix^-1I which is also ln I 1/x I.

Are my steps correct? If not, which step is incorrect?

Thank you.

Looks OK to me, but it's more wordy than it needs to be.
$-ln |x| = -1 * ln|x| = ln(|x|^{-1})$
The last expression is the same as $ln(\frac{1}{|x|})$