Why is eln x equal to x?

Answer 1

ln is an abbreviation of loge, that is logarithms taken to the base e.

The logarithm of a number (x) to a given base (b) is the number (y) such that b to the power y is the number (x), that is:

if y = logb x, then x = by.

So:

If y = loge x (= ln x) then x = ey

But y = ln x, thus x = eln x

by = x

Another way to see it (but the reason is almost identical to the above) is using the concept of inverse functions.

Say f is a function from set A to set B such that it is invertible (bijective), then there exist a function denoted f-1 such that for all f(x) = y,f-1(y) = x. Or f(f-1(x)) = f-1(f(x)) = x.

Now suppose this function f is from the real numbers to the real numbers defined as f(x) = ex. There is indeed a proof that this function is invertible, and we define the inverse to be f-1(y) = ln y. So in particular elnx = f(f-1(x)) = x = f-1(f(x)) = lnex.

As to the proof of why f(x) is indeed invertible, it involves the fact that is one-to-one and onto. Though straight forward, it is still not that trivial. I will leave the justification of why f is invertible to the reader's own research.

The definition of Ln(x) is

"the power to which 'e' must be raised in order to produce 'x'."

Now raise 'e' to that power:

e(the power to which 'e' must be raised in order to produce 'x')

and it should be pretty obvious that this operation produces 'x'.

Remember also that constants multiplied by logs can be rewritten as powers: e ln x = ln xe, which is more obviously x.

Answer 2

6

Answer 3

b