To rewrite \((\frac{a}{b})^2 \times c^3\) using the properties of logarithms, we can apply the logarithmic identities.
- Use the property \(\log(xy) = \log x + \log y\) for multiplication.
- Use the property \(\log(x^n) = n \log x\) for exponents.
First, consider \(y = (\frac{a}{b})^2 \times c^3\). Then the logarithm of \(y\) can be expressed as:
\[ \log y = \log\left((\frac{a}{b})^2 \times c^3\right) \]
Using the product rule of logarithms:
\[ \log y = \log\left((\frac{a}{b})^2\right) + \log(c^3) \]
Now, apply the power rule to each term:
\[ \log y = 2 \log\left(\frac{a}{b}\right) + 3 \log c \]
Next, we can further simplify \(\log\left(\frac{a}{b}\right)\) using the quotient rule:
\[ \log\left(\frac{a}{b}\right) = \log a - \log b \]
So, substituting this back, we have:
\[ \log y = 2(\log a - \log b) + 3 \log c \]
Distributing the 2 gives:
\[ \log y = 2 \log a - 2 \log b + 3 \log c \]
Thus, the final expression using the properties of logarithms is:
\[ \log y = 2 \log a - 2 \log b + 3 \log c \]