Question

If `log  (x + y)/2 = (log x + log y)/2`, prove that x = y.

Answer 1

Given: `log((x + y)/2) = (log x + log y)/2`, with x > 0, y > 0 log is defined.

To Prove: x = y

Proof [Step-wise]:

1. Start from the given equality:

`log((x + y)/2) = (log x + log y)/2`

2. Use log properties on the right-hand side:

`(log x + log y)/2`

= `(1/2) log (xy)` 

= `log ((xy)^(1/2))`

= `log (sqrt(xy))`

3. So we have `log ((x + y)/2) = log (sqrt(xy))`.

4. The logarithm for any base a with a > 0, a ≠ 1 is one-to-one on 0, ∞.

Hence, equal logs imply equal arguments `(x + y)/2 = sqrt(xy)`.

5. Multiply by 2:

`x + y = 2sqrt(xy)`

6. Square both sides:

(x + y)² = 4xy 

⇒ x² + 2xy + y² = 4xy

⇒ x² – 2xy + y² = 0

⇒ (x – y)² = 0

7. Therefore x – y = 0. so x = y.