Does ln[ln(x)] = ln(x) * ln(x)

[lilac30750]

Homework Statement

Does ln[ln(x) = ln(x) times ln(x)?

Homework Equations

The Attempt at a Solution

 

[Dick]

Well, no. Take x=e. ln(e)*ln(e)=1, ln(ln(e))=ln(1)=0.

 

[Mark44]

And generally speaking, f(f(x)) is different from f(x)*f(x). It's the difference between function composition (evaluation a function of a function) and function multiplication.

 

[Mentallic]

Maybe if we manipulated the expression a little it will become clearer that each function is not equal to each other.

$$ln\left(ln(x)\right)=\left(ln(x)\right)^2$$

$$ln(x)=e^{\left(ln(x)\right)^2}$$

$$ln(x)=(e^{ln(x)})^{ln(x)}$$

$$ln(x)=x^{ln(x)}$$

Does it look more obvious now as to why these expressions are not equal?