A weird relation for exponential function, is it really correct?

Asked Jun 18 '21 at 10:51

Active Jun 18 '21 at 11:04

Viewed 49 times

We written if $$f(x_1 + x_2) = f(x_1)f(x_2)$$ means that $f(x) = a^x$ and this is a characteristic property of the exponential functions.

This to me seems really weird, I am sure that there exist more functions for which this is true. Did the professor make a mistake here?

edited Jun 18 '21 at 11:04

asked Jun 18 '21 at 10:51

VLC

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Hmmm, do you mean ${f(x_1 + x_2) = f(x_1)f(x_2)}$? – Riemann'sPointyNose Jun 18 '21 at 10:53
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If he wrote it exactly like this, it is surely a typo... As it is, it's just false. – PierreCarre Jun 18 '21 at 10:59
@Riemann'sPointyNose Yes I did a typo, but the question is still the same – VLC Jun 18 '21 at 11:04
@PierreCarre Yes he written $\cdot$ instead of plus, but the question is still the same – VLC Jun 18 '21 at 11:05
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Why are you sure there are more functions satisfying this property? The result is actually correct. – PierreCarre Jun 18 '21 at 11:07
@PierreCarre I don't know it just seems weird to me, but if it is correct, than that answers my question – VLC Jun 18 '21 at 11:08
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If you take the logarithm on both sides then you get the Cauchy functional equation, which, under some mild conditions, has only linear functions as solutions. – Martin R Jun 18 '21 at 11:11
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@PierreCarre: If you don't require (e.g.) continuity then there are other solutions as well. – Martin R Jun 18 '21 at 11:15
@MartinR Yes, you are correct. – PierreCarre Jun 18 '21 at 11:17

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