260: Order of Powers

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A description/proof of order of powers

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Topics

About: arithmetic

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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

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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.

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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.

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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.

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Main Body

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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 $1<r$ or $r<1$, respectively.

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2: Proof 1

Let us admit that $e^x$ increases monotonously with respect to $x$. $r^x=(e^{lnr}{)}^x=e^{xlnr}$. When $1<r$, $0<lnr$, so, $xlnr$ increases monotonously when $x$ increases. When $r<1$, $lnr<0$, so, $xlnr$ decreases monotonously when $x$ increases.

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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 $0<r$ or $r<0$, respectively.

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4: Proof 2

Let us admit that $e^x$ increases monotonously with respect to $x$. $x^r=(e^{lnx}{)}^r=e^{rlnx}$. When $0<r$, $rlnx$ increases monotonously when $x$ increases. When $r<0$, $rlnx$ decreases monotonously when $x$ increases.

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References

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