Question

How do you find the inverse of `y=e^x`?

Answer 1

The inverse function is `lnx`.

Explanation:

By definition, `y=f^(-1)(x)ifff(y)=x`

`iffe^y=x`

`iffy=lnx`.

Answer 2

`x=ln(y)`

Explanation:

Given
`color(white)("XXX")y=e^x`

Taking the natural logarithm of both sides
`color(white)("XXX")color(red)(ln(y)=ln(e^x))`

By definition of `ln`
`color(white)("XXX")ln(a)=` the value. `b`, needed to make `e^b=a`
(you should memorize this)
therefore
`color(white)("XXX")ln(e^x)=` the value,`b`, needed to make `e^b=e^x`
that is
`color(white)("XXX")ln(e^x)=x`

Therefore `color(red)(ln(y)=ln(e^x))`
implies
`color(white)("XXX")color(blue)(x=ln(y))`