Question

If log (x + y) = log x – log y, prove that `x = y^2/(1 - y)`.

Answer 1

Given: log (x + y) = log x – log y, with logs in the same base and all logarithms defined so x > 0, y > 0, x + y > 0.

To Prove: `x = y^2/(1 - y)` with the necessary domain restriction on y.

Proof [Step-wise]:

1. Use the log subtraction rule:

`log x - log y = log (x/y)`

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

3. Since logs are equal same base and arguments positive

`x + y = x/y`

4. Multiply both sides by y:

y(x + y) = x

5. Expand:

xy + y² = x

6. Rearrange to isolate x:

y² = x – xy

= x(1 – y)

7. Solve for x:

`x = y^2/(1 - y)`

8. Check domain:

From logs we need x > 0 and x + y > 0. 

Substituting x gives `x + y = y/(1 - y)`. 

For this to be positive while y > 0 we require 1 – y > 0. 

So 0 < y < 1.

Also, y ≠ 1 denominator. 

Thus, the solution is valid for 0 < y < 1 and then x > 0 automatically.

`x = y^2/(1 - y)`, with 0 < y < 1 and y ≠ 1.