Find the positive value of x that satisfies the equation

Question

3log_{8^x=^​​​}​​​​​log₄^(x+6)

Answer 1

3 log₈x = log₄ (x+6)

⇒ 3 log_2³x = log_2² (x+6) (∵ log_a^mb = 1/m log_ab)

⇒ 3/3 log₂x = 1/2 log₂ (x+6)

⇒ 2 log₂x = log₂ (x+6)

⇒ log₂x² = log₂ (x+6)  (∵ n log a = log aⁿ)

⇒ x² = x+6  (By taking anti log)

⇒ x² - x - 6 = 0

⇒ x² - 3x + 2x - 6 = 0

⇒ x(x-3) + 2(x-3) = 0

⇒ (x+2) (x-3) = 0

⇒ x+2 = 0 or x-3 = 0

⇒ x = -2(Not possible) or x = 3)

(∵ Domain of log x is x > 0)

Hence, solution is x = 3