FACT
Okay, so I need to figure out the answer to the problem given. Let me start by understanding what's being asked here. The user mentioned "FACT" and then provided a problem where I have to determine the value of a certain expression involving logarithms. The expression is log base 2 of 7 multiplied by log base 7 of 2, and then that product is raised to the power of log base 2 of 3. Hmm, that sounds a bit complicated, but maybe there's a logarithm property that can simplify this.
First, I remember that there's a change of base formula for logarithms. The formula is log_b a = (log_c a) / (log_c b) for any positive number c ≠ 1. Maybe I can use that here. Let me look at the first part: log₂7 * log₇2. If I apply the change of base formula to one of these logarithms, maybe they will cancel each other out. Let's try converting log₇2 into log base 2. Using the change of base formula, log₇2 = (log₂2) / (log₂7). Since log₂2 is 1, this simplifies to 1 / log₂7. So then log₂7 * log₇2 becomes log₂7 * (1 / log₂7) = 1. Oh, that's nice! So the product of those two logarithms is just 1. That simplifies the original expression a lot.
Now the original expression is ^log₂3. Since we just found out that log₂7 * log₇2 = 1, this simplifies to 1^log₂3. But any number raised to any power is still 1, right? So 1 raised to the power of log₂3 is just 1. Therefore, the value of the expression should be 1. Let me double-check that to make sure I didn't make a mistake.
Starting again: log₂7 * log₇2. Let me verify using another approach. There's a logarithm identity that says log_b a * log_c b * log_a c = 1. Wait, but in this case, we have log₂7 * log₇2. If I set a = 7, b = 2, and c = 7, then log_b a * log_c b = log₂7 * log₇2. According to the identity, if we have a cycle of three logarithms, they multiply to 1, but here we only have two. However, if I consider log₂7 * log₇2, that's equivalent to (ln7 / ln2) * (ln2 / ln7) = 1. Yes, that's exactly what happens. The natural logs cancel out, leaving 1. So that confirms it again.
Therefore, the base of the exponent is 1, and 1 to any power is 1. So the final answer is 1. I think that's solid. Let me just make sure there's no alternative interpretation of the original problem. The expression is definitely ^log₂3. There's no ambiguity in the order of operations here because the multiplication is inside the brackets, and then the exponent is outside. So yeah, everything checks out. I don't see any mistakes in my reasoning.
https://www.18122.com/url/usdt/usdt0976.png
The value of the expression (\left( \log_2 7 \times \log_7 2 \right)^{\log_2 3}) is determined as follows:
Simplify the product of logarithms:
[
\log_2 7 \times \log_7 2
]
Using the change of base formula, (\log_b a = \frac{\log_c a}{\log_c b}), we rewrite (\log_7 2) as:
[
\log_7 2 = \frac{\log_2 2}{\log_2 7} = \frac{1}{\log_2 7}
]
Substituting back:
[
\log_2 7 \times \frac{1}{\log_2 7} = 1
]
Evaluate the exponent:
The original expression simplifies to:
[
1^{\log_2 3}
]
Since any number raised to any power is still 1:
[
1^{\log_2 3} = 1
]
Final Answer: (\boxed{1})
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