Question

If log₇ (P + Q) + log₇ (P² – PQ + Q²) = log₇ (14² - 7) and log₂ (P) + log₂ (Q) = log2  (20) then find value of P/Q
1. Either 4/9 or 9/4
2. 12/5
3. 13/5
4. Either 4/5 or 5/4

Answer 1

Correct Answer - Option 4 : Either 4/5 or 5/4

Calculation:

log₇ (P + Q) + log₇ (P² – PQ + Q²) = log₇ (196 – 7)

⇒ log₇ ((P + Q) × (P² – PQ + Q²)) = log₇ (189)

⇒ log₇ (P³ + Q³) = log₇ 189

⇒ P³ + Q³ = 189      ….(1)

log₂ P + log₂ Q = log2  20

⇒ log₂ (P × Q) = log2  20

⇒ P × Q = 20

⇒ P = 20/Q      ….(2)

Solving Equation 1 with help of Equation 2

⇒ (20/Q)³ + Q³ = 189

⇒ (8000/Q³) + Q³ = 189

⇒ 8000 + Q⁶ = 189Q³

⇒ Q⁶ - 189Q³ + 8000 = 0

Let, Q³ = X

⇒ X² – 189X + 8000

⇒ X = 64 or 125 both are right

Solving the equation,

⇒ Q = 5 and P = 4 or Q = 4 and P = 4

∴ P/Q = 4/5 or 5/4

log (X) + log (Y) = log (X × Y)  (This holds true only when bases of log under consideration are the same)