# Exercises: Exponential and Logarithm functions – Serlo

## Range of the exponential function

Exercise

Show that ${\displaystyle \exp ^{-1}\{0\}=\emptyset }$

You are supposed to show that ${\displaystyle \exp(z)\neq 0}$ holds for all ${\displaystyle z\in \mathbb {C} }$ .

How to get to the proof?

We have already shown that ${\displaystyle \exp(a)\neq 0}$ holds for all ${\displaystyle a\in \mathbb {R} }$. Let ${\displaystyle z}$ be an arbitrary complex number with ${\displaystyle z=a+ib}$, ${\displaystyle a,b\in \mathbb {C} }$.

Question: How can we express ${\displaystyle \exp(z)}$ using ${\displaystyle a}$ and ${\displaystyle b}$?

Using the exponential functional equation, we get

${\displaystyle \exp(z)=\exp(a+ib)=\exp(a)\exp(ib)}$

As ${\displaystyle \exp(a)\neq 0}$, ${\displaystyle \exp(z)\neq 0}$ holds if and only if ${\displaystyle \exp(ib)\neq 0}$. A good trick to show that a number is not equal to ${\displaystyle 0}$ is to show that its absolute value (or the square of its absolute value) is unequal to ${\displaystyle 0}$.

Question: What is ${\displaystyle |\exp(ib)|^{2}}$?

Using the computation rules for complex numbers, we get

${\displaystyle \left|\exp(ib)\right|^{2}=\exp(ib)\cdot {\overline {\exp(ib)}}=\exp(ib)\cdot \exp(-ib)=\exp(ib-ib)=\exp(0)=1}$

Hence, we have ${\displaystyle \exp(ib)\neq 0}$ and the claim follows.

Proof

We have already shown that ${\displaystyle \exp(a)\neq 0}$ holds for all ${\displaystyle a\in \mathbb {R} }$. Furthermore, for all ${\displaystyle b\in \mathbb {R} }$, holds that ${\displaystyle \left|\exp(ib)\right|^{2}=\exp(ib)\cdot {\overline {\exp(ib)}}=\exp(ib)\cdot \exp(-ib)=\exp(ib-ib)=\exp(0)=1}$. Hence, for all ${\displaystyle b\in \mathbb {R} }$ follows that ${\displaystyle \exp(ib)\neq 0}$.

Let ${\displaystyle z}$ be an arbitrary complex number. Then there exist ${\displaystyle a,b\in \mathbb {R} }$ such that ${\displaystyle z=a+ib}$. It follows that ${\displaystyle \exp(z)=\exp(a+ib)=\underbrace {\exp(a)} _{\neq 0}\cdot \underbrace {\exp(ib)} _{\neq 0}\neq 0}$.