Question

Solve for x:

log (x + 3) + log (x – 5) = 2 log 3

Answer 1

We are given:

log(x + 3) + log(x – 5) = 2 log 3

Step 1: Combine the left-hand side

log[(x + 3)(x – 5)]

Right-hand side:

2 log 3 = log(3²) = log 9

So the equation becomes:

log[(x + 3)(x – 5)] = log 9

Step 2: Equate the arguments

(x + 3)(x – 5) = 9

Expand:

x² – 5x + 3x – 15 = 9

⇒ x² – 2x – 15 = 9

⇒ x² – 2x – 24 = 0

Step 3: Solve the quadratic

x² – 2x – 24 = 0

⇒ (x – 6)(x + 4) = 0

⇒ x = 6 or x = – 4

Step 4: Check validity

x = 6 ⇒ x + 3 = 9, x – 5 = 1 → valid

x = – 4 ⇒ x + 3 = –1 → invalid (log of negative)

x = 6