Question

Expanding Logarithmic Expressions In Exercise, use the properties of logarithms to rewrite the expression as the sum, difference, or multiple of logarithms.
In x^2/x^2 + 1

Answer 1

Answer:

$\\ ln(x^2) - ln(x^2 + 1)$.

For values of x>0, it can be rewritten as $\\ 2ln(x) - ln(x^2 + 1).$

Step-by-step explanation:

For the expression:

$\\ ln(\frac{x^2}{x^2 + 1})$

We can apply this logarithmical property: $\\ ln(\frac{x}{y}) = ln(x) - ln(y).$

Then,

$\\ \frac{x^2}{x^2 + 1} = ln(x^2) - ln(x^2 + 1).$

If we assume values of x > 0 (non negative values for x), then the expression could be rewritten as follows:

$\\ \frac{x^2}{x^2 + 1} = ln(x^2) - ln(x^2 + 1) = 2 ln(x) - ln(x^2 + 1)$, since $\\ ln(x^{n}) = n*ln(x).$

We have to remember that domain (all possible values x) for logarithmic function is for all x > 0, or mathematically expressed as:

Domain: $\\ {x | x \in R, x > 0}$