Logarithmic Equation Resolution

Logarithmic Properties and Equations

Key Concepts

• Logarithmic Identity:
  • Logarithms allow the combination of logarithmic terms using properties of logarithms.
• Equality:
  • Establishing equality through the use of logarithmic identities is critical in solving logarithmic equations.

Detailed Breakdown of the Equation

• The given equation is:
  $\log(3x + 2) + \log(x - 1) = 1$

Applying Logarithmic Properties

• Sum of Logarithms:

  • The sum of two logarithms can be combined into a single logarithm:
    $\log(a) + \log(b) = \log(ab)$
  • Applying this property:
    $\log(3x + 2) + \log(x - 1) = \log((3x + 2)(x - 1))$

• Therefore, we can rewrite the equation as:
  $\log((3x + 2)(x - 1)) = 1$

Converting from Logarithmic to Exponential Form

• Exponential Form:
  • To solve for x, we convert the logarithmic equation into its exponential form:
    $a = \log_b(c) \implies b^a = c$
  • Here, base 10 is implicit. Hence:
    $10^1 = (3x + 2)(x - 1)$
  • Which simplifies to:
    $10 = (3x + 2)(x - 1)$

Final Steps to Solve for x

• Expand and Rearrange:

  • Expanding the right side:
    $10 = 3x^2 - 3x + 2x - 2$
  • Simplifies to:
    $10 = 3x^2 - x - 2$

• Setting Up the Quadratic Equation:

  • Rearranging gives:
    $3x^2 - x - 12 = 0$

• Solving the Quadratic:

  • Apply the quadratic formula:
    $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
  • Where:
  • a = 3, b = -1, c = -12

• Calculating:
  $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(3)(-12)}}{2(3)}$

  • Leads to:
    $x = \frac{1 \pm \sqrt{1 + 144}}{6}$
  • Further simplifying gives:
    $x = \frac{1 \pm 13}{6}$

Possible Solutions

• The two potential solutions derived are:
  1. $x = \frac{14}{6} = \frac{7}{3}$
  2. $x = \frac{-12}{6} = -2$

Verifying Solutions

• Since logarithms are not defined for negative inputs or zero, check which solution is valid:
  • For $x = \frac{7}{3}$:
  • Both $3x + 2$ and $x - 1$ are positive.
  • For $x = -2$:
  • This is invalid because x - 1 = -3 < 0.

Conclusion

• Valid Solution:
  • The only acceptable solution to the equation is:
    $x = \frac{7}{3}$