Question

log(x + 3) + log(x – 3) = 4 log 2.

∴ The value of x is ______.

Options

• 4

• 5

• 6

• 7

Answer 1

∴ The value of x is 5.

Explanation:

log(x + 3) + log(x – 3) = 4 log 2

Step 1: Apply the logarithmic rule:

log a + log b = log (ab)

So, log(x + 3)(x – 3) = 4 log 2

Step 2: Simplify:

(x + 3)(x − 3)

= x² − 9

log(x² – 9) = log (2)⁴

Step 3: Since logs are equal, their arguments are equal.

x² – 9 = 16

Step 4: Solve for x:

x² = 25

x = ±5

Step 5: Check the validity:

For x = 5, both x + 3 = 8 and x – 3 = 2 are positive, so it is valid.

For x = –5, both x + 3 = –2 and x – 3 = –8 are negative, which makes the logarithms undefined.

Therefore, the valid solution is x = 5.