Solve for X: Log₈(4x) + Log₈(x+2) = 3Log₈(3) Equation

To solve the given equation $\frac{\log_8(4x) + \log_8(x+2)}{\log_8 3} = 3$, we follow these steps:

• Step 1: Combine the logs in the numerator using the product rule

  We use the product rule: $\log_8(4x) + \log_8(x+2) = \log_8((4x)(x+2)) = \log_8(4x^2 + 8x)$.

• Step 2: Equate the fraction to 3 and solve the resulting equation

  This gives us $\frac{\log_8(4x^2 + 8x)}{\log_8 3} = 3$.

  Cross-multiplying, we have $\log_8(4x^2 + 8x) = 3\log_8 3$.

  By the power rule, we can simplify as $\log_8(4x^2 + 8x) = \log_8 3^3 = \log_8 27$.

• Step 3: Solve for $x$

  Since the logarithms are the same base, we equate the arguments: $4x^2 + 8x = 27$.

  Rearranging gives the quadratic equation $4x^2 + 8x - 27 = 0$.

  We solve this quadratic equation using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a = 4$, $b = 8$, and $c = -27$.

  Thus, $x = \frac{-8 \pm \sqrt{8^2 - 4 \cdot 4 \cdot (-27)}}{2 \cdot 4}$.

  Calculating further, $x = \frac{-8 \pm \sqrt{64 + 432}}{8}$.

  This simplifies to $x = \frac{-8 \pm \sqrt{496}}{8}$.

  Simplifying $\sqrt{496} = 4\sqrt{31}$, the equation becomes:

  $x = \frac{-8 \pm 4\sqrt{31}}{8}$.

  Further simplifying gives us two solutions: $x = -1 \pm \frac{\sqrt{31}}{2}$.

Given that $x$ must be positive for the original logarithms to be valid, we take $x = -1 + \frac{\sqrt{31}}{2}$.

Therefore, the correct solution is $x = -1+\frac{\sqrt{31}}{2}$.