Question

If \( f(x) = \log \left(\frac{(1 + x)}{(1 - x)}\right) \), then \( f(x) + f(y) \) is:

• A. \( f(x + y) \)
• B. \( \frac{(x + y)}{(1 + xy)} \)
• C. \( \frac{1}{(1 + xy)} \)
• D. \( f(x) + f(y) \)

Hint:

When dealing with logarithms, always remember to use the logarithmic identity \( \log a + \log b = \log(ab) \).

Answer 1

The Correct Option is B

Using the given function and properties of logarithms: \[ f(x) + f(y) = \log \left(\frac{(1+x)}{(1-x)}\right) + \log \left(\frac{(1+y)}{(1-y)}\right) \] Using the logarithmic property \( \log a + \log b = \log(ab) \), we get: \[ f(x) + f(y) = \log \left( \frac{(1+x)(1+y)}{(1-x)(1-y)} \right) \] Expanding the terms gives: \[ f(x) + f(y) = \log \left( \frac{(x+y)}{(1+xy)} \right) \] Thus, the Correct Answer is \( \frac{(x+y)}{(1+xy)} \).