Math for Non-Geeks/ Exercises: Exponential and Logarithm functions

oben Range of the exponential function Show that You are supposed to show that holds for all . We have already shown that holds for all . Let be an arbitrary complex number with , . 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 . What is ? Using the computation rules for complex numbers, we get Hence, we have and the claim follows. 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 . unten