Question

Solve for x:

log (x – 6) + log (x – 3) = 2 log 2

Answer 1

We are given:

log(x – 6) + log(x – 3) = 2 log 2

Step 1: Use log properties

Left side:

log[(x – 6)(x – 3)]

Right side:

2 log 2 = log 2² = log 4

Now the equation is:

log[(x – 6)(x – 3)] = log 4

Step 2: Equate arguments

(x – 6)(x – 3) = 4

Expand:

x² – 9x + 18 = 14

⇒ x² – 9x + 14 = 0

Step 3: Solve the quadratic

x² – 9x + 14 = 0

⇒ (x – 7)(x – 2) = 0

⇒ x = 7 or x = 2

Step 4: Check validity

x = 7 ⇒ x – 6 = 1, x – 3 = 4 → valid

x = 2 ⇒ x – 6 = –4 → invalid (log of negative)

x = 7