Find the solution. Select the correct next step. \( \log _{4}(8 x)-\log _{4}(x-7)=3 \) \( \log _{4}\left(\frac{8 x}{x-7}\right)=3 \quad \log _{4}(8 x(x-7))=3 \quad \log _{4}\left(\frac{x-7}{8 x}\right)=3 \)

To solve the equation \( \log _{4}(8 x)-\log _{4}(x-7)=3 \), we can use the properties of logarithms. The left-hand side can be simplified using the quotient rule for logarithms, which states that \( \log_a(b) - \log_a(c) = \log_a\left(\frac{b}{c}\right) \). Therefore, the correct next step is \( \log _{4}\left(\frac{8 x}{x-7}\right)=3 \). Once you have that, you can convert the logarithmic equation into its exponential form. Since \( \log _{4}\left(\frac{8 x}{x-7}\right)=3 \) means that \( \frac{8 x}{x-7} = 4^3 = 64 \). From there, you can solve for \( x \) by cross-multiplying and rearranging the equation. Now, when working with logarithmic equations, it's crucial to check for extraneous solutions. Remember that the argument of a logarithm must be greater than zero. After solving for \( x \), you must verify that \( x - 7 > 0 \) and \( 8x > 0 \) to ensure that all parts of the logarithm are valid!