Question
Compare the two logarithms by placing < , > , or = in the response box.(1 point)
Answers
Answer
Compare the two logarithms by placing < , > , or = in the response box.(1 point) log200 log8/ ln200ln8
Answered by
GPT-4o mini
Sure! Please provide the two logarithmic expressions you'd like to compare, and I'll help you determine their relationship.
Answered by
GPT-4o mini
To compare the two logarithmic expressions \( \frac{\log 200}{\log 8} \) and \( \frac{\ln 200}{\ln 8} \), we can use the change of base formula for logarithms.
The change of base formula states that for any positive numbers \( a, b \), and base \( c \), the following holds:
\[
\log_a b = \frac{\log_c b}{\log_c a}
\]
Let’s express both logarithms using a common base, say natural logarithm (ln):
1. We have:
\[
\log 200 = \frac{\ln 200}{\ln 10}
\]
\[
\log 8 = \frac{\ln 8}{\ln 10}
\]
Therefore:
\[
\frac{\log 200}{\log 8} = \frac{\frac{\ln 200}{\ln 10}}{\frac{\ln 8}{\ln 10}} = \frac{\ln 200}{\ln 8}
\]
2. So we notice that:
\[
\frac{\log 200}{\log 8} = \frac{\ln 200}{\ln 8}
\]
Thus, we can conclude:
\[
\frac{\log 200}{\log 8} = \frac{\ln 200}{\ln 8}
\]
**Final answer**: \( = \)
The change of base formula states that for any positive numbers \( a, b \), and base \( c \), the following holds:
\[
\log_a b = \frac{\log_c b}{\log_c a}
\]
Let’s express both logarithms using a common base, say natural logarithm (ln):
1. We have:
\[
\log 200 = \frac{\ln 200}{\ln 10}
\]
\[
\log 8 = \frac{\ln 8}{\ln 10}
\]
Therefore:
\[
\frac{\log 200}{\log 8} = \frac{\frac{\ln 200}{\ln 10}}{\frac{\ln 8}{\ln 10}} = \frac{\ln 200}{\ln 8}
\]
2. So we notice that:
\[
\frac{\log 200}{\log 8} = \frac{\ln 200}{\ln 8}
\]
Thus, we can conclude:
\[
\frac{\log 200}{\log 8} = \frac{\ln 200}{\ln 8}
\]
**Final answer**: \( = \)
Submit Your Answer
We prioritize human answers over AI answers.
If you are human, and you can answer this question, please submit your answer.