X to the x is not of Exponential Order/Lemma
Jump to navigation
Jump to search
Lemma
Let $f: \R_{>0} \to \R$ be defined as:
- $\forall x \in \R_{>0}: \map f x = x^x$
Let there exist strictly positive real constants $M, K, a \in \R_{> 0}$ such that:
- $\forall t \ge M: \size {\map f t} < K e^{a t}$
Then there exists a constant $C$ such that:
- $\forall t > C: \size {\map f t} > K e^{a t}$
Proof
By the definition of power:
- $\map f t = \map \exp {t \ln t}$
By Exponential of Real Number is Strictly Positive, we can reduce the lemma into the existence of $C$ such that:
- $\forall t > C: \map f t > K e^{a t}$
We will divide into two cases.
Case 1: $K > 1$
Assume that $t > K e^a$.
| \(\ds a\) | \(>\) | \(\ds 0\) | ||||||||||||
| \(\ds e^a\) | \(>\) | \(\ds e^0\) | Exponential is Strictly Increasing | |||||||||||
| \(\ds e^a\) | \(>\) | \(\ds 1\) | Exponential of Zero | |||||||||||
| \(\ds K e^a\) | \(>\) | \(\ds 1\) | As $K > 1$ | |||||||||||
| \(\ds t\) | \(>\) | \(\ds 1\) | As $t > K e^a$ | |||||||||||
| \(\text {(1)}: \quad\) | \(\ds t \ln K\) | \(>\) | \(\ds \ln K\) | |||||||||||
| \(\ds t\) | \(>\) | \(\ds K e^a\) | Assumption | |||||||||||
| \(\ds \ln t\) | \(>\) | \(\ds \map \ln {K e^a}\) | Logarithm is Strictly Increasing | |||||||||||
| \(\ds \ln t\) | \(>\) | \(\ds a + \ln K\) | ||||||||||||
| \(\ds t \ln t\) | \(>\) | \(\ds a t + t \ln K\) | ||||||||||||
| \(\ds t \ln t\) | \(>\) | \(\ds a t + \ln K\) | from $(1)$ | |||||||||||
| \(\ds \map \exp {t \ln t}\) | \(>\) | \(\ds \map \exp {a t + \ln K}\) | Exponential is Strictly Increasing | |||||||||||
| \(\ds \map f t\) | \(>\) | \(\ds K e^{a t}\) | Exponential is Strictly Increasing |
Here, $C = K e^a$.
Case 2: $K \le 1$
Assume that $t > e^a$.
| \(\ds \ln K\) | \(\le\) | \(\ds \ln 1\) | Logarithm is Strictly Increasing | |||||||||||
| \(\text {(1)}: \quad\) | \(\ds \ln K\) | \(\le\) | \(\ds 0\) | Logarithm of 1 is 0 | ||||||||||
| \(\ds t\) | \(>\) | \(\ds e^a\) | by assumption | |||||||||||
| \(\ds \ln t\) | \(>\) | \(\ds a\) | Logarithm is Strictly Increasing | |||||||||||
| \(\ds t \ln t\) | \(>\) | \(\ds a t\) | ||||||||||||
| \(\ds t \ln t\) | \(>\) | \(\ds a t + \ln k\) | from $(1)$ | |||||||||||
| \(\ds \map \exp {t \ln t}\) | \(>\) | \(\ds \map \exp {a t + \ln K}\) | Exponential is Strictly Increasing | |||||||||||
| \(\ds \map f t\) | \(>\) | \(\ds K e^{a t}\) | Exponential is Strictly Increasing |
Here:
- $C = e^a$
$\blacksquare$