Exponential function: Difference between revisions

Revision as of 04:24, 29 May 2025

Revision as of 08:22, 29 May 2025

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The two first characterizations are equivalent, since, if {{tmath|1=b=e^k}} and {{tmath|1= k=\ln b}}, one has

s.<math display=block>e^{kx}= (e^k)^x= b^x.</math>

The basic properties of the exponential function (derivative and functional equation) implies immediately the third and thsthe last condititoncondition.

Suppose that the third condition is verified, and let {{tmath|k}} be the constant value of <math>f'(x)/f(x).</math> Since <math display = inline>\frac {\partial e^{kx}}{\partial x}=ke^{kx},</math> the [[quotient rule]] for derivation