Question
Which of the following is equal to log2 8^4/3?
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GPT-4o mini
To solve \(\log_2 \left(\frac{8^4}{3}\right)\), we can start by breaking it down using the properties of logarithms.
1. Use the property of logarithms that states \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\):
\[
\log_2 \left(\frac{8^4}{3}\right) = \log_2 (8^4) - \log_2 (3)
\]
2. Next, simplify \(\log_2 (8^4)\). Since \(8 = 2^3\), we can express \(8^4\) as \((2^3)^4 = 2^{12}\). Then we apply the logarithm:
\[
\log_2 (8^4) = \log_2 (2^{12}) = 12
\]
3. Now, substitute back into the logarithm expression we derived earlier:
\[
\log_2 \left(\frac{8^4}{3}\right) = 12 - \log_2 (3)
\]
So the final result is:
\[
\log_2 \left(\frac{8^4}{3}\right) = 12 - \log_2 (3)
\]
This expression simplifies our original logarithmic expression. If you just need the value as is, it remains:
\[
\log_2 \left(\frac{8^4}{3}\right) = 12 - \log_2 (3)
\]
1. Use the property of logarithms that states \(\log_b \left(\frac{m}{n}\right) = \log_b m - \log_b n\):
\[
\log_2 \left(\frac{8^4}{3}\right) = \log_2 (8^4) - \log_2 (3)
\]
2. Next, simplify \(\log_2 (8^4)\). Since \(8 = 2^3\), we can express \(8^4\) as \((2^3)^4 = 2^{12}\). Then we apply the logarithm:
\[
\log_2 (8^4) = \log_2 (2^{12}) = 12
\]
3. Now, substitute back into the logarithm expression we derived earlier:
\[
\log_2 \left(\frac{8^4}{3}\right) = 12 - \log_2 (3)
\]
So the final result is:
\[
\log_2 \left(\frac{8^4}{3}\right) = 12 - \log_2 (3)
\]
This expression simplifies our original logarithmic expression. If you just need the value as is, it remains:
\[
\log_2 \left(\frac{8^4}{3}\right) = 12 - \log_2 (3)
\]
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