Question

Evaluate x if log₃ (3 + x) + log₃ (8 – x) – log₃ (9x – 8) = 2 – log₃9

(a) 2 

(b) –2 

(c) 4 

(d) –4

Answer 1

(c) 4

log₃ (3 + x) + log₃ (8 – x) – log₃ (9x – 8) = 2 – log₃9 

⇒ log₃ (3 + x) + log₃ (8 – x) – log₃ (9x – 8) + log₃9 = 2

⇒ log3 \(\bigg[\frac{(3+x)(8-x)(9)}{(9x-8)}\bigg]=2\)

⇒ \(\frac{9(24+8x-3x-x^2)}{(9x-8)}=3^2=9\)

⇒ –x² + 5x + 24 = 9x – 8 ⇒ x² + 4x – 32 = 0 

⇒ (x + 8) (x – 4) = 0 ⇒ x = – 8, 4. 

Taking the positive value x = 4.