To rewrite the logarithmic equation without logarithms, we start with the given equation:
\[ 3 \log x = \log 64 \]
Using properties of logarithms, we can first rewrite \( 3 \log x \) as \( \log(x^3) \). Thus, we get:
\[ \log(x^3) = \log 64 \]
Since the logarithmic functions are equal, we can set their arguments equal to each other (as long as the arguments are in their domain, which we'll check later):
\[ x^3 = 64 \]
Now, we can solve for \( x \) by taking the cube root:
\[ x = 64^{1/3} = 4 \]
Next, we need to check the solution to ensure it is within the domain of the original logarithmic expressions. The argument of the logarithm, \( x \), must be greater than zero (\( x > 0 \)). Since our solution \( x = 4 \) is greater than zero, it is valid.
Now we can express the solution set:
The solution set is \({ 4 }\).
Final answer:
A. The solution set is \( { 4 } \).