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Proposition: General Powers of Positive Numbers

Let \(x\) be any real number and let \(a > 0 \) be a positive real number. Then the general power function \(a\to a^x\) is well-defined and equals the exponential function of general base \(a\), formally:

\[a^x=\exp_a(x).\]

The following interactive figure demonstrates the general power in relation to different values of exponents \(x\in[-20,20]\).

Table of Contents

Proofs: 1

1. Proposition: Calculation Rules for General Powers
2. Proposition: Derivative of General Powers of Positive Numbers
3. Proposition: Integral of General Powers

Mentioned in:

Corollaries: 1
Lemmas: 2
Proofs: 3 4 5
Propositions: 6 7 8 9 10 11 12 13

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References

Bibliography

1. Forster Otto: "Analysis 1, Differential- und Integralrechnung einer Veränderlichen", Vieweg Studium, 1983