Question

Question #95ade

Answer 1

Another way of expressing the natural logarithm of some number `a` is, "what power do I have to raise `e` to to get `a`?", or more formally solving this equation for `y`:
`e^y=a`.

Let's do the same for this expression:
`y=ln(e^x)`

`e^y=e^x`

It is quite apparent from this that `x=y`, but if we want to do it more formally, we can take the natural log of both sides:
`cancel(ln)(cancel(e)^y)=cancel(ln)(cancel(e)^x)`

`y=x`

Since `y=ln(e^x)`, we have just shown that it is equal to `x`:
`ln(e^x)=x`

Answer 2

Acceptable proof will depend on your definitions of `lnx` and `e^x`

Explanation:

One treatment defines `lnx = int_1^x 1/t dt`

Then defines `e^x` to be the number whose `ln` is `x`.

In this treatment, it is immediate from the definition that `ln(e^x) = x`.