To determine which of the provided expressions is equivalent to [tex]\(\log \left(\frac{20}{3}\right)\)[/tex], we can use the properties of logarithms, particularly the property of logarithms concerning the division of numbers.
The logarithmic property used here is:
[tex]\[
\log \left( \frac{a}{b} \right) = \log(a) - \log(b)
\][/tex]
Given the logarithm expression [tex]\(\log \left(\frac{20}{3}\right)\)[/tex], we can apply the property:
[tex]\[
\log \left( \frac{20}{3} \right) = \log(20) - \log(3)
\][/tex]
Now, let's examine the options:
A. [tex]\( 20 \cdot \log(3) \)[/tex]
This option implies a multiplication rather than an appropriate logarithmic operation. It does not match our derived expression.
B. [tex]\(\log (20) \cdot \log (3)\)[/tex]
This option also involves multiplication of two logarithm expressions, which does not align with the division rule of logarithms.
C. [tex]\(\log(20) + \log(3)\)[/tex]
This option suggests the addition of logarithms. However, the correct property adds the logarithms when we are dealing with a product inside the logarithm, not a quotient.
[tex]\[
\log(ab) = \log(a) + \log(b)
\][/tex]
So, this does not match our requirement either.
D. [tex]\(\log (20) - \log (3)\)[/tex]
This option correctly represents the division property of logarithms and matches the expression [tex]\(\log \left( \frac{20}{3} \right) = \log(20) - \log(3)\)[/tex].
Thus, the correct expression equivalent to [tex]\(\log \left(\frac{20}{3}\right)\)[/tex] is:
[tex]\[
\boxed{\text{D.} \log (20) - \log (3)}
\][/tex]
