Answer:
[tex] \\ ln(x^2) - ln(x^2 + 1) [/tex].
For values of x>0, it can be rewritten as [tex] \\ 2ln(x) - ln(x^2 + 1). [/tex]
Step-by-step explanation:
For the expression:
[tex] \\ ln(\frac{x^2}{x^2 + 1}) [/tex]
We can apply this logarithmical property: [tex] \\ ln(\frac{x}{y}) = ln(x) - ln(y). [/tex]
Then,
[tex] \\ \frac{x^2}{x^2 + 1} = ln(x^2) - ln(x^2 + 1). [/tex]
If we assume values of x > 0 (non negative values for x), then the expression could be rewritten as follows:
[tex] \\ \frac{x^2}{x^2 + 1} = ln(x^2) - ln(x^2 + 1) = 2 ln(x) - ln(x^2 + 1) [/tex], since [tex] \\ ln(x^{n}) = n*ln(x). [/tex]
We have to remember that domain (all possible values x) for logarithmic function is for all x > 0, or mathematically expressed as:
Domain: [tex] \\ {x | x \in R, x > 0} [/tex]
