Solve the logarithmic equation log₄ x + logₓ 16 = 3, x > 0, x ≠ 1.

Step 1: Convert logₓ 16 using change of base
logₓ 16 = log₄ 16 / log₄ x
Since 16 = 4²: log₄ 16 = 2
Therefore: logₓ 16 = 2/log₄ x

Step 2: Let y = log₄ x
Equation becomes: y + 2/y = 3

Step 3: Multiply by y
y² + 2 = 3y
y² – 3y + 2 = 0

Step 4: Factorize
(y – 1)(y – 2) = 0
y = 1 or y = 2

Step 5: Convert back to x
If y = 1: log₄ x = 1 → x = 4¹ = 4
If y = 2: log₄ x = 2 → x = 4² = 16
Both solutions are valid