To solve the equation
\[ \log_{81}(32x) - \log_{81}(x - 3) = \frac{3}{4}, \]
we can use the properties of logarithms, specifically that the subtraction of logarithms can be rewritten as a logarithm of a quotient:
\[ \log_{81}\left(\frac{32x}{x - 3}\right) = \frac{3}{4}. \]
We can eliminate the logarithm by exponentiating both sides with base 81:
\[ \frac{32x}{x - 3} = 81^{\frac{3}{4}}. \]
Calculating \(81^{\frac{3}{4}}\):
Since \(81 = 3^4\), we have:
\[ 81^{\frac{3}{4}} = (3^4)^{\frac{3}{4}} = 3^3 = 27. \]
Now we replace \(81^{\frac{3}{4}}\) in our equation:
\[ \frac{32x}{x - 3} = 27. \]
Next, we can cross-multiply:
\[ 32x = 27(x - 3). \]
Expanding the right side gives:
\[ 32x = 27x - 81. \]
Now, we can isolate \(x\):
\[ 32x - 27x = -81, \] \[ 5x = -81. \] \[ x = -\frac{81}{5}. \]
Thus, the solution for \(x\) is
\[ x = -\frac{81}{5}. \]
Therefore, the correct answer from the provided responses is:
x equals start fraction negative 81 over 5 end fraction.