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The equation you've provided, \(\log_8(x) = a \log_2(x)\), seems to suggest a relationship between the logarithms with bases 8 and 2. To analyze this, let's apply the change of base formula:

\[
\log_b(x) = \frac{\log_k(x)}{\log_k(b)}
\]

For your equation, we use base \(k = 2\) (since \(2\) is a factor of \(8\)), and this gives:

\[
\log_8(x) = \frac{\log_2(x)}{\log_2(8)}
\]

Since \(\log_2(8) = 3\) (because \(2^3 = 8\)), we have:

\[
\log_8(x) = \frac{\log_2(x)}{3}
\]

Comparing this to your original equation \(\log_8(x) = a \log_2(x)\), we can equate:

\[
\frac{\log_2(x)}{3} = a \log_2(x)
\]

If \(x\) is not equal to \(1\) (since \(\log_2(1) = 0\) would make both sides zero), we can simplify to find \(a\):

\[
\frac{1}{3} = a
\]

Thus, the correct formulation should be:

\[
\log_8(x) = \frac{1}{3} \log_2(x)
\]

Do you have any questions or need further details on this topic? Here are some additional questions to consider:
1. How does the change of base formula apply to logarithms with other bases?
2. Can you solve for \(x\) if given a specific value for \(\log_8(x)\) or \(\log_2(x)\)?
3. What properties of logarithms are useful in simplifying expressions involving multiple logarithmic terms?
4. How would this change if we considered logarithms with a base other than \(2\) or \(8\)?
5. How does the logarithm function behave graphically across different bases?

Tip: Remember, the change of base formula is a powerful tool that can help convert complex logarithmic expressions into simpler ones, especially when dealing with bases that are powers of each other.

Learn how to solve the logarithmic equation log_8(x) = a log_2(x) using the change of base formula. This problem involves converting between different logarithmic bases and solving for coefficients. Suitable for students in Grades 9-12.

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