Question

What is the value of `ln e^4`?

Answer 1

`ln e^4= color(red)4`

Explanation:

Remember
`color(white)("XXX")ln color(blue)a = color(magenta)b`
means
`color(white)("XXX")e^color(magenta)b=color(blue)a`

So if `ln color(blue)(e^4) = color(magenta)b`
then
`color(white)("XXX")e^color(magenta)b=color(blue)(e^4)`

So
`color(white)("XXX")color(magenta)b=4`
and
`color(white)("XXX")lncolor(blue)(e^4)color(white)("xx")[=color(magenta)b]color(white)("xx")=4`

Answer 2

`4`

Explanation:

The logarithm of a power is the power `xx` the logarithm of the number.

Example:

`lna^n => nlne`

So:
`lne^4 => 4lne`

The logarithm of the base is always `1`

Proof:

`lne = y=> e^y = e => e^y = e^1`(if bases are the same then powers are equal)

Example:

`log_10(10)= 1`

`log_2(2)=1`

So `lne=1`

So we have:

`4lne=> 4(1)=4`