Question

If log₂ x + log₂ y ≥ 6, then the least value of x + y is ______.

Options

• 4

• 8

• 16

• 32

Answer 1

If log₂ x + log₂ y ≥ 6, then the least value of x + y is 16.

Explanation:

Given, log₂ x + log₂ y ≥ 6

`\implies` log₂ (xy) ≥ 6

`\implies` xy ≥ 2⁶

`\implies sqrt(xy) ≥ 2^3`

∵ `(x + y)/2 ≥ sqrt(xy)` or `x + y ≥ 2sqrt(xy) ≥ 16`  ...[∵ AM ≥ GM]

∴ x + y ≥ 16