To rewrite [tex]\(3 \log x + 4 \log (x-2)\)[/tex] as a single logarithm, we can use the properties of logarithms. Let's break this down step-by-step:
1. Power Rule: The power rule of logarithms states that [tex]\( k \log(a) = \log(a^k) \)[/tex]. This rule allows us to bring the coefficients inside the logarithm as exponents.
Applying the power rule to each term:
- [tex]\(3 \log x = \log(x^3)\)[/tex]
- [tex]\(4 \log (x-2) = \log((x-2)^4)\)[/tex]
2. Product Rule: The product rule of logarithms states that [tex]\( \log(a) + \log(b) = \log(a \cdot b) \)[/tex]. This rule allows us to combine the two logarithms into a single logarithm.
Applying the product rule to the transformed terms:
- [tex]\(\log(x^3) + \log((x-2)^4) = \log(x^3 \cdot (x-2)^4)\)[/tex]
Thus, [tex]\(3 \log x + 4 \log (x-2)\)[/tex] can be written as a single logarithm: [tex]\(\log(x^3 (x-2)^4)\)[/tex].
Therefore, the best answer from the choices provided is:
a. [tex]\(\log x^3(x-2)^4\)[/tex]
So, the final answer is:
A
