Solve for \( x \) : \( \log _{5} x+\log _{5}(x+3)=3 \) \( x=\square \) Question Help: Video Submit Question

To solve the equation \( \log _{5} x+\log _{5}(x+3)=3 \), we can use the property of logarithms that allows us to combine them: \[ \log_{5}(x(x+3)) = 3. \] By rewriting this in exponential form, we get: \[ x(x + 3) = 5^3, \] which simplifies to: \[ x(x + 3) = 125. \] This leads to the quadratic equation: \[ x^2 + 3x - 125 = 0. \] Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) with \( a = 1, b = 3, c = -125 \): \[ x = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 1 \cdot (-125)}}{2 \cdot 1} = \frac{-3 \pm \sqrt{9 + 500}}{2} = \frac{-3 \pm \sqrt{509}}{2}. \] Calculating \( \sqrt{509} \) gives us approximately 22.6. Therefore, \[ x = \frac{-3 + 22.6}{2} \quad \text{(we ignore the negative root since \( x \) must be positive)}, \] which results in: \[ x \approx \frac{19.6}{2} \approx 9.8. \] Thus, rounding \( x \) gives us \( x \approx 9.8 \), or more neatly, we express the solution as \( x \approx \frac{-3 + \sqrt{509}}{2} \). So, \( x \approx \sqrt{509} / 2 - 3/2 \). Answer: \( x \approx 9.8 \) \( x=\sqrt{509}/2 - 3/2 \).