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In Math and Arithmetic

# Why is eln x equal to x?

ln is an abbreviation of log e , 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 t…he number (x), that is: if y = log b x, then x = b y. So: If y = log e x (= ln x) then x = e y But y = ln x, thus x = e ln x .
b y = 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) = e x . 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 e lnx = f(f -1 (x)) = x = f -1 (f(x)) = lne x . 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 x e , which is more obviously x.. ( Full Answer )