Question

If 4x2+9y2=16 xy, then show that 2 log (2x-3y)=2 log 2+log x+logy.​

Answer 1

2 log(2x - 3y) = 2 log2 + logx + logy

Step-by-step explanation:

Given,

4x² + 9y² = 16xy

or, (2x)² + (3y)² = 16xy

or, (2x)² - (2 * 2x * 3y) + (3y)² = 16xy - (2 * 2x * 3y)

or, (2x - 3y)² = 16xy - 12xy

or, (2x - 3y)² = 4xy

Taking log in both sides, we get

log{(2x - 3y)²} = log(4xy)

or, 2 log(2x - 3y) = log4 + logx + logy

or, 2 log(2x - 3y) = log(2²) + logx + logy

or, 2 log(2x - 3y) = 2 log2 + logx + logy

Hence proved.

Rules:

• log(ab) = loga + logb

• log(aᵇ) = b loga

• (a - b)² = a² - 2ab + b²