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Definition: Exponential Function of General Base

Let \(a > 0\) be a positive real number, and let \(\ln(a)\) be the natural logarithm of \(a\). The exponential function of general base \(a\) is defined as the exponential function:

\[a^x:=\exp_a(x):=\begin{cases}\mathbb R&\mapsto \mathbb R\\x&\mapsto \exp(x\cdot \ln(a)).\end{cases}\]

Table of Contents

1. Proposition: Continuity of Exponential Function of General Base
2. Proposition: Functional Equation of the Exponential Function of General Base
3. Proposition: Exponential Function of General Base With Natural Exponents
4. Proposition: Exponential Function of General Base With Integer Exponents
5. Proposition: Functional Equation of the Exponential Function of General Base (Revised)

Mentioned in:

Corollaries: 1
Proofs: 2 3 4 5 6 7 8
Propositions: 9 10 11 12 13 14 15 16

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References

Bibliography

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