The question asks to identify which of the listed expressions are equivalent to $\log_8 20$.

Using the Change of Base Formula:

The change of base formula for logarithms is: $\log_b a = \frac{\log_c a}{\log_c b}$ where $b$ is the base and $a$ is the argument. Applying this to $\log_8 20$, we can express it as:

$\log_8 20 = \frac{\log 20}{\log 8}$ This means that $\frac{\log 20}{\log 8}$ is an equivalent expression to $\log_8 20$, which corresponds to the second option.

Evaluating the Other Options:

• The first option, $\frac{\log 8}{\log 20}$, is not correct because it is the reciprocal of $\log_8 20$, not equivalent.
• The third option, $\frac{\log_{10} 20}{\log_{10} 8}$, is equivalent to $\frac{\log 20}{\log 8}$ because the change of base formula can be applied to any base, including base 10 (common logarithm). So, this is also correct.
• The fourth option, $\log_{20} 8$, is not equivalent because the base and argument are reversed.

Correct Answers:

• $\frac{\log 20}{\log 8}$
• $\frac{\log_{10} 20}{\log_{10} 8}$

Would you like more details or have any questions? Here are some related questions to explore:

1. How does the change of base formula work for logarithms?
2. What happens if you switch the base and the argument in a logarithmic expression?
3. How do you simplify logarithmic expressions involving different bases?
4. Can logarithms with different bases ever be equivalent to one another?
5. What are common mistakes to avoid when using the change of base formula?

Tip: When simplifying logarithmic expressions, always consider applying the change of base formula for clarity.