Question

Question #3acdb

Answer 1

If x = y + h, where h is relatively small in magnitude, then `x*ln(x/y) = x-y`, nearly, neglecting terms of higher order smallness in `h^2, h^3, h^4, ...`

Explanation:

Let x = y + h, where h is relatively small in magnitude.

`x*ln(x/y) = (y + h)* ln((y + h)/y) = (y + h)*ln(1 + h/y) = (y + h)(h/y-(1/2)(h^2/y^2)+(1/3)(h^3/y^3-...)`, using series expansion for `ln(1+(h/y))` in power of `(h/y)`.

This leads to
`x*ln(x/y)` = h + terms of order higher powers of h
= h + `O(h^2)`
= x-y, nearly..