Question

How do you simplify `ln e^(2x)`?

Answer 1

`ln e^(2x) = 2x`

Explanation:

As a Real valued function, `x |-> e^x` is one to one from `(-oo, oo)` onto `(0, oo)`.

As a result, for any `y in (0, oo)` there is a unique Real value `ln y` such that `e^(ln y) = y`.

This is the definition of the Real natural logarithm.

If `t in (-oo, oo)` then `y = e^t in (0, oo)` and from the above definition:

`e^(ln(e^t)) = e^t`

Since `x |-> e^x` is one to one, we can deduce that for any Real value of `t`:

`ln e^t = t`

In other words, `t |-> e^t` and `t |-> ln t` are mutual inverses as Real valued functions.

So if `x` is any Real value:

`ln e^(2x) = 2x`

Answer 2

`2x`

Explanation:

Using the property of logs:

`log(a^b) = b log a`

We can see that:

`ln(e^(2x))=2x ln e`

And since `ln(e) = log_e(e)=1`,

`2xlne=2x`

Answer 3

`2x`

Explanation:

The key realization here is that `lnx` and `e^x` are inverses of each other, which cancel each other out. So we essentially have

`cancel(ln)cancel(e)^(2x)`

which just leaves us with `color(blue)(2x)`.

Hope this helps!