If `log_(a)x = p` and `log_(b)x^(2) =q,` then `log_(x)sqrt(ab)` is equal to (where, a,b, `x in R^(+)-{1})-`
A) `1/p+1/q`, B) `1/(2p)+1/q`, C) `1/p+1/(2q)`, D) `1/(2p)+1/(2q)`

Explanation

To solve the problem, we need to find the value of \( \log_{x} \sqrt{ab} \) given that \( \log_{a} x = p \) and \( \log_{b} x^{2} = q \).

### Step-by-step Solution:

1. **Convert the logarithmic expressions:**
- From \( \log_{a} x = p \), we can express this in terms of base \( x \):
\[
\log_{x} a = \frac{1}{p}
...