To expand the logarithmic expression $\log(3x^2)$ fully using properties of logarithms, we can proceed as follows:

1. Apply the Product Rule of Logarithms: The product rule states that $\log(ab) = \log(a) + \log(b)$. $\log(3x^2) = \log(3) + \log(x^2)$

2. Apply the Power Rule of Logarithms: The power rule states that $\log(a^b) = b \cdot \log(a)$. $\log(3) + \log(x^2) = \log(3) + 2 \cdot \log(x)$

So, the fully expanded expression is: $\log(3x^2) = \log(3) + 2 \log(x)$

This is the final answer, expressed in terms of $\log(x)$.

Would you like more details on any step, or have any questions?

Here are five related questions to deepen understanding:

1. How would you expand $\log(5x^3)$ using the properties of logarithms?
2. What is the expanded form of $\log(2x^4y^3)$?
3. If $\log(3) \approx 0.477$ and $\log(x) = 1$, what is the numerical value of $\log(3x^2)$?
4. How do the properties of logarithms apply when dividing terms, for example with $\log\left(\frac{x^2}{5}\right)$?
5. What is the purpose of using the product and power rules when working with logarithmic expressions?

Tip: Remember that breaking down logarithmic expressions using these properties helps simplify complex calculations, especially when variables or constants have exponents.