document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=464116>Question 464116</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Calculate all of the solutions of the following equation:<BR>\n" );
document.write( "e^(-z+5) = 2i </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #318207 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=318207\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Note that\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE 2i = 2e^{i\frac{\pi}{2}} = e^{\ln{2}} e^{i\frac{\pi}{2}} = e^{\ln{2} + i\frac{\pi}{2}}\">\r<BR>\n" );
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document.write( "Hence, all we need to solve is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{-z + 5} = e^{\ln{2} + i\frac{\pi}{2}}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE -z + 5 = \ln{2} + i\frac{\pi}{2}\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE z = 5 - \ln{2} - i\frac{\pi}{2}\">\r<BR>\n" );
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document.write( "Note that we can replace pi/2 with 5pi/2, 9pi/2, ... because e^{ix} is periodic with period 2pi. The general solution is\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE z = 5 - \ln{2} - i(2n\pi + \frac{\pi}{2})\"> where n is any integer. </SPAN>\n" );
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