To solve the equation:

$6 \log_2 x = -\log_2 64$

Step 1: Simplify the Right Side

The term $\log_2 64$ can be simplified. Since $64 = 2^6$, we have:

$\log_2 64 = \log_2 (2^6) = 6$

Thus, the equation becomes:

$6 \log_2 x = -6$

Step 2: Divide Both Sides by 6

To isolate $\log_2 x$, divide both sides by 6:

$\log_2 x = -1$

Step 3: Rewrite in Exponential Form

Now, rewrite the logarithmic equation $\log_2 x = -1$ in exponential form:

$x = 2^{-1}$

Step 4: Simplify

Calculating $2^{-1}$ gives:

$x = \frac{1}{2}$

Solution

The solution set is:

$\{ \frac{1}{2} \}$

Final Answer

The correct choice is:

• A. The solution set is $\{ \frac{1}{2} \}$.

Would you like more details on the solution or have any questions?

1. What are the properties of logarithms that can simplify similar equations?
2. How can you convert between logarithmic and exponential forms?
3. What would the solution be if the base of the logarithm was different?
4. How do you handle equations involving multiple logarithmic terms?
5. What are other common types of logarithmic equations?

Tip: Always check if a logarithmic argument results in a negative value, as logarithms are undefined for negative numbers in real numbers.