Let V = C {z < 0}. Define the principal logarithm Log: V → C as he branch of log on V such that Log(1) = 0. (a) Prove that such a branch of log exists on V. (b) Prove for z E V that Log(z) = ln |z| + iArg(2), where In is the natural logarithm on positive reals, and Arg is the principal argument with image (-n, 7]. (c) Is Log(zw) = Log(z) + Log(w) for all z, w e V? Justify your answer. Similarly, is Log(z") = nLog(z) for all n E Z and z E V? Justify your answer.

Let V = C {z < 0}. Define the principal logarithm Log: V → C as he branch of log on V such that Log(1) = 0. (a) Prove that such a branch of log exists on V. (b) Prove for z E V that Log(z) = ln |z| + iArg(2), where In is the natural logarithm on positive reals, and Arg is the principal argument with image (-n, 7]. (c) Is Log(zw) = Log(z) + Log(w) for all z, w e V? Justify your answer. Similarly, is Log(z") = nLog(z) for all n E Z and z E V? Justify your answer.

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.6: Exponential And Logarithmic Equations

Problem 20E

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