Solving Logarithmic Equation

Equation Given:

• (16^{\log_4(x-2)} + 10\log(x+1) - 4\log_4(5) = 0)

Key Concepts and Steps:

Properties of Logarithms:

• Basic Understanding:

  • Example: (7^{\log_7 9} = 9)
  • Reasoning: (a^{\log_a b} = b), using properties of logs and exponents.

• Change of Base Formula:

  • (\log_a b = \frac{\log b}{\log a})

Solving the Equation:

1. Understanding Base:

   • If no base is shown, assume base 10.
   • (10^{\log(x+1)} = x+1)

2. Converting (16^{\log_4(x-2)}):

   • Recognize that (16 = 4^2).
   • Rewrite: (16^{\log_4(x-2)} = (4^2)^{\log_4(x-2)} = 4^{2\cdot\log_4(x-2)}).
   • Use property: (a^{m\cdot n} = (a^m)^n).

3. Simplify Expression:

   • Using (4^{\log_4 k} = k):
     • (4^{\log_4(x-2)} = x-2).
   • Therefore, (16^{\log_4(x-2)} = (x-2)^2).

4. Simplified Expression:

   • ((x-2)^2 + (x+1) - 5 = 0)

5. FOIL Method for ((x-2)^2):

   • (x^2 - 2x - 2x + 4).
   • Simplify: (x^2 - 4x + 4 - 4 = 0) becomes (x^2 - 3x = 0).

6. Factor the Equation:

   • (x(x - 3) = 0).

7. Solutions: x = 0 or x = 3

   • Check if allowed: Cannot have log of a negative number or zero.
     • (x = 0) is invalid as it results in log(-2).
   • Validate (x = 3):
     • Substitute and simplify to ensure the equation holds.
     • Results show both sides equal, confirming (x = 3) is valid.

Conclusion:

• Solution: The only valid solution is (x = 3).