Question

How do you evaluate `ln(ln e^(e^100))`?

Answer 1

100

Explanation:

`ln (e^a)=a`
The bracketed logarithm `ln( (e)^(e^100))=e^100`
The given expression = `ln (e^100)` = 100.

Answer 2

`100`

Explanation:

We have:

`ln(ln(e^(e^100)))`

Within the innermost logarithm, we can use the following rule:

`ln(color(blue)a^color(red)b)=color(red)b*ln(color(blue)a)`

This gives us:

`ln(ln(color(blue)e^(color(red)(e^100))))=ln(color(red)(e^100)*ln(color(blue)(e)))`

Since `ln(e)=1`, this equals

`ln(e^100*ln(e))=ln(e^100)`

Using the previously defined exponent rule, we can rewrite this as follows:

`ln(color(blue)e^color(red)100)=color(red)100*ln(color(blue)e)=barul|color(white)(a/a)100color(white)(a/a)|`