Simplify 3 ln(𝑥𝑦2) − ln(𝑥𝑦)2

Sure, let's simplify the given expression step by step.

Given expression: 3 ln(𝑥𝑦²) − ln(𝑥𝑦)²

Step 1: Apply the power rule of logarithms, which states that log_b(m^n) = n * log_b(m), to both terms:

= 3 * [ln(x) + 2ln(y)] - 2 * [ln(x) + ln(y)]

Step 2: Distribute the constants:

= 3ln(x) + 6ln(y) - 2ln(x) - 2ln(y)

Step 3: Combine like terms:

= (3ln(x) - 2ln(x)) + (6ln(y) - 2ln(y))

= ln(x) + 4ln(y)

So, 3 ln(𝑥𝑦²) − ln(𝑥𝑦)² simplifies to ln(x) + 4ln(y).