Question

Find the value of x in the following:

log (x + 2) + log (x – 2) = log 2 + log 3 + 1

Answer 1

Given: log (x + 2) + log (x – 2) = log 2 + log 3 + 1, with log understood as base 10 so 1 = log 10. 

Step-wise calculation:

1. Domain:

x – 2 > 0 

⇒ x > 2

2. Combine left logs:

log [(x + 2)(x – 2)]

= log (x² – 4)

3. Combine right logs:

log 2 + log 3 + 1

= log (2 × 3) + log 10 

= log (6) + log (10) 

= log (60)

4. Equate:

log (x² – 4) = log (60)

⇒ x² – 4 = 60

⇒ x² = 64

⇒ x = ±8

5. Apply domain x > 2

⇒ Discard x = −8

⇒ x = 8