Proposition: If z and w are nonzero complex numbers, then log zw = log z + log w. (a) How is this equality to be interpreted? (b) Why is it necessary that we specify how this equality is to be interpreted? (c) How do we know that ln |zw| + i arg zw = ln |z| + i arg z + ln |w| + i arg w?

Proposition: If z and w are nonzero complex numbers, then log zw = log z + log w. (a) How is this equality to be interpreted? (b) Why is it necessary that we specify how this equality is to be interpreted? (c) How do we know that ln |zw| + i arg zw = ln |z| + i arg z + ln |w| + i arg w?

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.3: The Addition And Subtraction Formulas

Problem 62E

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Question

answer the questions which follow:
Proposition: If z and w are nonzero complex numbers, then log zw = log z + log w.
(a) How is this equality to be interpreted?
(b) Why is it necessary that we specify how this equality is to be interpreted?
(c) How do we know that ln |zw| + i arg zw = ln |z| + i arg z + ln |w| + i arg w?
(d) Is it always true that Log zw = Log z + Log w?

Combination of a real number and an imaginary number. They are numbers of the form a + b , where a and b are real numbers and i is an imaginary unit. Complex numbers are an extended idea of one-dimensional number line to two-dimensional complex plane.

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