To expand the logarithmic expression [tex]\(\log \left(\frac{7 \cdot y^3}{4 \cdot x^2}\right)\)[/tex], we can use several properties of logarithms. Let's go through the steps in detail:
1. Identify the properties of logarithms:
- [tex]\(\log \left(\frac{a}{b}\right) = \log a - \log b\)[/tex] (Logarithm of a quotient)
- [tex]\(\log (a \cdot b) = \log a + \log b\)[/tex] (Logarithm of a product)
- [tex]\(\log (a^b) = b \log a\)[/tex] (Logarithm of a power)
2. Apply the quotient property:
We start with the given expression:
[tex]\[
\log \left(\frac{7 \cdot y^3}{4 \cdot x^2}\right)
\][/tex]
Using the quotient property, we get:
[tex]\[
\log \left(\frac{7 \cdot y^3}{4 \cdot x^2}\right) = \log (7 \cdot y^3) - \log (4 \cdot x^2)
\][/tex]
3. Apply the product property to both the numerator and the denominator separately:
For the numerator [tex]\(\log (7 \cdot y^3)\)[/tex]:
[tex]\[
\log (7 \cdot y^3) = \log 7 + \log y^3
\][/tex]
For the denominator [tex]\(\log (4 \cdot x^2)\)[/tex]:
[tex]\[
\log (4 \cdot x^2) = \log 4 + \log x^2
\][/tex]
4. Substituting back into the expression:
[tex]\[
\log \left(\frac{7 \cdot y^3}{4 \cdot x^2}\right) = (\log 7 + \log y^3) - (\log 4 + \log x^2)
\][/tex]
5. Apply the power property to [tex]\(\log y^3\)[/tex] and [tex]\(\log x^2\)[/tex]:
[tex]\[
\log y^3 = 3 \log y
\][/tex]
[tex]\[
\log x^2 = 2 \log x
\][/tex]
6. Combine all the terms:
[tex]\[
\log \left(\frac{7 \cdot y^3}{4 \cdot x^2}\right) = \log 7 + \log y^3 - \log 4 - \log x^2
\][/tex]
Using the power property:
[tex]\[
\log 7 + 3 \log y - \log 4 - 2 \log x
\][/tex]
So, the correctly expanded form of the logarithmic expression [tex]\(\log \left(\frac{7 \cdot y^3}{4 \cdot x^2}\right)\)[/tex] is:
[tex]\[
\boxed{\log 7 + 3 \log y - \log 4 - 2 \log x}
\][/tex]
