Question

Lne^x=5 I know that x=5 but why?

Answer 1

The function `e^x:RR->(0, oo)` and `ln:(0, oo)->RR` are inverse functions of one another.

So for any `x in RR`, `ln(e^x) = x` and for any `x in (0, oo)`, `e^ln(x) = x`

Explanation:

The natural logarithm `ln(x)` is defined such that `e^ln(x) = x`

So what about `ln(e^x)` ?

By the definition of `ln`, we have `e^(ln(e^x)) = e^x`

Since `e^x` is a one-one function, we can deduce that the exponents `ln(e^x)` and `x` must be equal.

That is:

`ln(e^x) = x`