Question

Find the value of x in the following:

log (x + 4) + log (x – 4) = 4 log 2 + log 3

Answer 1

Given: log (x + 4) + log (x – 4) = 4 log 2 + log 3

Step-wise calculation:

1. Use log a + log b = log (ab): 

log [(x + 4)(x – 4)] = 4 log 2 + log 3

2. Combine the right-hand logs:

4 log 2 + log 3

= log (2⁴) + log 3 

= log (16) + log 3 

= log (16 × 3) 

= log 48

3. Therefore

log [(x + 4)(x – 4)] = log 48 

⇒ (x + 4)(x – 4) = 48

4. Expand:

x² – 16 = 48 

⇒ x² = 64

⇒ x = ±8

5. Domain check:

Arguments of logs require x + 4 > 0 and x – 4 > 0.

So x > 4.

This excludes x = –8.

⇒ x = 8