Question

Solve for x:

log (4x – 3) + log (4x + 3) = 4 log 2

Answer 1

log (4x – 3) + log (4x + 3) = 4 log 2

Step 1: Use the logarithmic property:

log a + log b = log (ab)

log [(4x – 3)(4x + 3)] = 4 log 2

Simplify the left-hand side:

log [(4x – 3)(4x + 3)] = log (16x² – 9)

Step 2: Simplify the right-hand side

The right-hand side is 4 log 2 and using the logarithmic property a log b = log(b^a), we get:

4 log 2 = log (2⁴) = log 16

So now the equation becomes:

log (16x² – 9) = log 16

Step 3: Equate the arguments of the logarithms:

Since log a = log b implies a = b, we get:

16x² – 9 = 16

Step 4: Solve for x:

16x² = 16 + 9 = 25

`x^2 = 25/16 = (5/4)^2`

`x = +-5/4`

Step 5: Check validity:

For `x = 5/4`, both 4x – 3 = 5 and 4x + 3 = 8, so the logarithms are valid.

For `x = -5/4`, both 4x – 3 = –8 and 4x + 3 = 0, which makes the logarithms undefined.

Thus, the only valid solution is `x = 5/4`.