To analyze whether Joy or Payne is correct in their rewrites of \( \log_5{20} \), we first need to recall the change of base formula for logarithms. The change of base formula states:
\[ \log_a{b} = \frac{\log_c{b}}{\log_c{a}} \]
for any positive base \( c \).
Let's see how each person rewrote \( \log_5{20} \).
Joy's Rewrite: Joy rewrote \( \log_5{20} \) as \( \frac{\log{20}}{\log{5}} \). Here, Joy is using the logarithm with base 10 (common logarithm). According to the change of base formula, this is valid:
\[ \log_5{20} = \frac{\log_{10}{20}}{\log_{10}{5}} = \frac{\log{20}}{\log{5}} \]
So Joy's representation is correct.
Payne's Rewrite: Payne rewrote \( \log_5{20} \) as \( \frac{\ln{20}}{\ln{5}} \). Here, Payne is using the natural logarithm (base \( e \)). The change of base formula also applies here:
\[ \log_5{20} = \frac{\ln{20}}{\ln{5}} \]
So Payne's representation is also correct.
Conclusion:
Both rewrites are correct. Joy and Payne both used valid forms of the change of base formula, but with different bases for their logarithms. Therefore, the answer is both are correct.