Score: 4.86/8 Answered: 7/8 Question 8 Simplify. Enter the result as a single logarithm with a coefficient of 1. log_(8)(6x)-log_(8)(7x^7)=square

To simplify the expression \(\log_{8}(6x) - \log_{8}(7x^{7})\), we can use the properties of logarithms. Specifically, we will use the quotient rule for logarithms, which states:<br /><br />\[<br />\log_b(a) - \log_b(c) = \log_b\left(\frac{a}{c}\right)<br />\]<br /><br />Applying this property to the given expression:<br /><br />\[<br />\log_{8}(6x) - \log_{8}(7x^{7}) = \log_{8}\left(\frac{6x}{7x^{7}}\right)<br />\]<br /><br />Next, we simplify the fraction inside the logarithm:<br /><br />\[<br />\frac{6x}{7x^{7}} = \frac{6}{7} \cdot \frac{x}{x^{7}} = \frac{6}{7} \cdot x^{1-7} = \frac{6}{7} \cdot x^{-6}<br />\]<br /><br />So the expression becomes:<br /><br />\[<br />\log_{8}\left(\frac{6}{7} \cdot x^{-6}\right)<br />\]<br /><br />Using the product rule for logarithms, which states \(\log_b(ab) = \log_b(a) + \log_b(b)\), we can further simplify:<br /><br />\[<br />\log_{8}\left(\frac{6}{7} \cdot x^{-6}\right) = \log_{8}\left(\frac{6}{7}\right) + \log_{8}(x^{-6})<br />\]<br /><br />Since \(\log_{8}(x^{-6}) = -6 \log_{8}(x)\), we have:<br /><br />\[<br />\log_{8}\left(\frac{6}{7}\right) + \log_{8}(x^{-6}) = \log_{8}\left(\frac{6}{7}\right) - 6 \log_{8}(x)<br />\]<br /><br />Thus, the simplified form of the original expression is:<br /><br />\[<br />\boxed{\log_{8}\left(\frac{6}{7x^{6}}\right)}<br />\]