Logarithmic and Exponential Equations Score: 5.5/10 Answered: 7/10 Solve and Check: log_(3)(x)+log_(3)(x+8)=2 If there is a solution, enter as an integer or reduced fraction If there are multiple solutions, separate them using commas. For no solutions, click No solution. One or more solutions: square No solution

To solve the equation $log_{3}(x)+log_{3}(x+8)=2$, we can use the properties of logarithms to combine the two logarithmic terms on the left side of the equation.<br /><br />Using the product rule of logarithms, we have:<br /><br />$log_{3}(x) + log_{3}(x+8) = log_{3}(x(x+8))$<br /><br />So the equation becomes:<br /><br />$log_{3}(x(x+8)) = 2$<br /><br />Now, we can use the definition of a logarithm to rewrite the equation in exponential form:<br /><br />$3^2 = x(x+8)$<br /><br />Simplifying the equation, we get:<br /><br />$9 = x^2 + 8x$<br /><br />Rearranging the terms, we have:<br /><br />$x^2 + 8x - 9 = 0$<br /><br />This is a quadratic equation, which can be solved using the quadratic formula:<br /><br />$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$<br /><br />In this case, $a = 1$, $b = 8$, and $c = -9$. Plugging these values into the quadratic formula, we get:<br /><br />$x = \frac{-8 \pm \sqrt{8^2 - 4(1)(-9)}}{2(1)}$<br /><br />$x = \frac{-8 \pm \sqrt{64 + 36}}{2}$<br /><br />$x = \frac{-8 \pm \sqrt{100}}{2}$<br /><br />$x = \frac{-8 \pm 10}{2}$<br /><br />So the solutions are:<br /><br />$x_1 = \frac{-8 + 10}{2} = 1$<br /><br />$x_2 = \frac{-8 - 10}{2} = -9$<br /><br />Therefore, the solutions to the equation $log_{3}(x)+log_{3}(x+8)=2$ are $x = 1$ and $x = -9$.