Question

Solve for x:

log (5x + 6) + log (5x – 6) = 2 log 8

Answer 1

We are given:

log(5x + 6) + log(5x – 6) = 2 log 8

Step 1: Use log product rule

log[(5x + 6)(5x – 6)] = 2 log 8

Simplify both sides:

• Left: (5x + 6)(5x – 6) = 25x² – 36
• Right: 2 log 8 = log 8² = log 64

So the equation becomes:

log(25x² – 36) = log 64

Step 2: Equate the arguments

25x² – 36 = 64

25x² = 100

x² = 4

⇒ x = ±2

Step 3: Check validity

• For x = 2:
  5x + 6 = 16 > 0, 5x – 6 = 4 > 0 → valid
• For x = –2:
  5x + 6 = –10 + 6 = – 4 < 0 → invalid

x = 2