f(x) = ln(ln(ln(x))); find the domain of f?

Question

How is this problem solved?

Answer 1

f(x) = ln(ln(ln(x)))

We know what the restriction of just ln(x) is; it is such that x > 0.

Therefore, it follows that

ln(ln(x)) > 0

If we take the antilog of both sides (that is, bring both sides to the power of e), we will get rid of the ln, to get

ln(x) > e^0, or
ln(x) > 1. Again, we take the antilog of both sides of the inequality, to obtain

x > e^1, or
x > e

Therefore, since this inequality represents the values which x CAN be, it also represents the domain.

The domain of f is the interval (e, infinity).

Note that e has a round bracket because the inequality is strictly greater than.

Answer 2

You want to take the ln four times so my guess would be e^e
to infinity, so that the outermost ln would have a postive arguemnt.
Let's see how that checks out...
ln(ln(ln(ln(x)))
ln(ln(ln(ln((e^e) =
ln(ln(ln(e)=
ln(ln(1) =
ln(0) =
So the domain would be (e^e, infinity). You wouldn't include the endpoint e^e because in the final step you would be taking ln 0, which is undefined. But anything greater than e^e, would be ok in tihs function.

Answer 3

ln (x) is defined when x > 0.

So, f(x) is defined when ln(ln(x)) > 0
i.e. when exp(ln(ln(x))) > exp(0)
i.e. when ln (x) > 1
i.e. when x > exp(1) = e.
So the domain is (e, infinity).

Answer 4

log e to the y
domain -1 to 1
i am just taking a guess