Solve the logarithmic equation: log₂ (6 – x) = 3 – log₂ x

Explanation

Step 1: Rearrange equation
log₂(6 – x) = 3 – log₂ x
log₂(6 – x) + log₂ x = 3

Step 2: Apply log product rule
log₂(6 – x) + log₂ x = log₂[(6 – x) × x]
log₂[x(6 – x)] = 3

Step 3: Convert to exponential form
If log₂(y) = 3, then y = 2³
x(6 – x) = 8

Step 4: Expand and rearrange
6x – x² = 8
-x² + 6x – 8 = 0
x² – 6x + 8 = 0

Step 5: Factorize
x² – 6x + 8 = 0
(x – 4)(x – 2) = 0
x = 4 or x = 2

Step 6: Verify both solutions
For x = 4: log₂(2) = 3 – log₂(4) → 1 = 3 – 2 ✓
For x = 2: log₂(4) = 3 – log₂(2) → 2 = 3 – 1 ✓
Both are valid