Question 9 Solve: log_(4)(-10x+7)-log_(4)(-6x-18)=2 x=square (Enter DNE if no solution exists)

To solve the equation, we can use the properties of logarithms to combine the two logarithmic expressions on the left side of the equation:<br /><br />$log_{4}(-10x+7)-log_{4}(-6x-18)=2$<br /><br />Using the quotient rule for logarithms, we can rewrite the equation as:<br /><br />$log_{4}\left(\frac{-10x+7}{-6x-18}\right)=2$<br /><br />Now, we can use the definition of a logarithm to rewrite the equation in exponential form:<br /><br />$\frac{-10x+7}{-6x-18}=4^2$<br /><br />Simplifying the right side of the equation:<br /><br />$\frac{-10x+7}{-6x-18}=16$<br /><br />Cross-multiplying:<br /><br />$-10x+7=16(-6x-18)$<br /><br />Expanding the right side of the equation:<br /><br />$-10x+7=-96x-288$<br /><br />Combining like terms:<br /><br />$86x=-295$<br /><br />Dividing both sides by 86:<br /><br />$x=-\frac{295}{86}$<br /><br />Therefore, the solution to the equation is $x=-\frac{295}{86}$.