Question

solve for (x):

\\\log_{2} x + \log_{2} (x + 3) = 7\\

(x =)

Explanation:

Step1: Apply product rule

$$\log_{2}(x(x + 3)) = 7$$

Step2: Rewrite in exponential form

$$x(x + 3) = 2^7$$

Step3: Simplify the equation

$$x^2 + 3x - 128 = 0$$

Step4: Apply quadratic formula

$$x = \frac{-3 \pm \sqrt{3^2 - 4(1)(-128)}}{2(1)}$$

Step5: Simplify the discriminant

$$x = \frac{-3 \pm \sqrt{521}}{2}$$

Step6: Check domain constraints

$$x = \frac{-3 + \sqrt{521}}{2}$$

Answer:

\(x = \frac{-3 + \sqrt{521}}{2}\)