# Exercises: Exponential and Logarithm functions – Serlo

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• Introduction • Complex numbers • Supremum and infimum • Sequences • Convergence and divergence • Subsequences, Accumulation points and Cauchy sequences • Series • Convergence criteria for series • Exponential and Logarithm functions • Derivation and definition of the exponential series • Properties of the exponential series • Logarithmic function • Real exponents • Exp and log functions for complex numbers • Exercises • Trigonometric and Hyperbolic functions • Continuity • Differential Calculus ## Range of the exponential function

Exercise

Show that $\exp ^{-1}\{0\}=\emptyset$ You are supposed to show that $\exp(z)\neq 0$ holds for all $z\in \mathbb {C}$ .

How to get to the proof?

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

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

Using the exponential functional equation, we get

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

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

Using the computation rules for complex numbers, we get

$\left|\exp(ib)\right|^{2}=\exp(ib)\cdot {\overline {\exp(ib)}}=\exp(ib)\cdot \exp(-ib)=\exp(ib-ib)=\exp(0)=1$ Hence, we have $\exp(ib)\neq 0$ and the claim follows.

Proof

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

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