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.