Exercises: Exponential and Logarithm functions – Serlo

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Range of the exponential function[Bearbeiten]

Exercise

Show that

You are supposed to show that holds for all .

How to get to the proof?

We have already shown that holds for all . Let be an arbitrary complex number with , .

Question: How can we express using and ?

Using the exponential functional equation, we get

As , holds if and only if . A good trick to show that a number is not equal to is to show that its absolute value (or the square of its absolute value) is unequal to .

Question: What is ?

Using the computation rules for complex numbers, we get

Hence, we have and the claim follows.

Proof

We have already shown that holds for all . Furthermore, for all , holds that . Hence, for all follows that .

Let be an arbitrary complex number. Then there exist such that . It follows that .