Question

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

Answer 1

Given: log (x + y) = log x + log y

To Prove: `y = x/(x - 1)`

Proof [Step-wise]:

1. By the product rule for logarithms

log x + log y = log (xy)

2. Hence log (x + y) = log (xy).

3. The logarithm is one-to-one on its domain of positive arguments 

So x + y = xy with x > 0, y > 0.

4. Rearrange:

xy – y = x

5. Factor:

y(x – 1) = x

6. Solve for y:

`y = x/(x - 1)`

7. Note domain restrictions:

x ≠ 1 otherwise denominator 0.

If x = 1 the original equality leads to a contradiction.

So x ≠ 1; also x > 0, y > 0.

So the logarithms are defined.

Therefore, from log (x + y) = log x + log y with x, y > 0, we obtain `y = x/(x - 1)` and x ≠ 1.