A description/proof of order of powers
Topics
About: arithmetic
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Description 1
- 2: Proof 1
- 3: Description 2
- 4: Proof 2
Starting Context
- The reader knows a definition of power.
- The reader admits the proposition that the power of \(e\) to an exponent increases monotonously with respect to the exponent.
Target Context
- The reader will have a description and a proof of the proposition that the power of any positive base to an exponent increases or decreases monotonously with respect to the exponent when the base is larger than \(1\) or is smaller than \(1\), respectively; the power of a positive base to any exponent increases or decreases monotonously with respect to the base when the exponent is larger than \(0\) or is smaller than \(0\), respectively.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Description 1
For any positive real number, \(r\), and any real number, \(x\), \(r^x\) increases or decreases monotonously with respect to \(x\) when \(1 \lt r\) or \(r \lt 1\), respectively.
2: Proof 1
Let us admit that \(e^x\) increases monotonously with respect to \(x\). \(r^x = (e^{ln r})^x = e^{x ln r}\). When \(1 \lt r\), \(0 \lt ln r\), so, \(x ln r\) increases monotonously when \(x\) increases. When \(r \lt 1\), \(ln r \lt 0\), so, \(x ln r\) decreases monotonously when \(x\) increases.
3: Description 2
For any real number, \(r\), and any positive real number, \(x\), \(x^r\) increases or decreases monotonously with respect to \(x\) when \(0 \lt r\) or \(r \lt 0\), respectively.
4: Proof 2
Let us admit that \(e^x\) increases monotonously with respect to \(x\). \(x^r = (e^{ln x})^r = e^{r ln x}\). When \(0 \lt r\), \(r ln x\) increases monotonously when \(x\) increases. When \(r \lt 0\), \(r ln x\) decreases monotonously when \(x\) increases.
