document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=568663>Question 568663</A>: <I><SPAN STYLE=\"word-break: break-all;\"> I'm not entirely sure this is the right subject, sorry if it's not...\r<BR>\n" );
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document.write( "A substance decomposes radioactively. Its half-life is 32 years. Find the constant \"k\" in the decay formula.<BR>\n" );
document.write( "Use this formula instead: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=ne%5Ekt ALIGN=MIDDLE  ALT=\"y=ne%5Ekt\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "y= final value<BR>\n" );
document.write( "n= initial value<BR>\n" );
document.write( "k= constant<BR>\n" );
document.write( "t= time<BR>\n" );
document.write( "e= antilogarithm  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #367064 by </B><A HREF=/tutors/aboutme.mpl?userid=solver91311><B>solver91311(24713)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=solver91311><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=solver91311><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=367064\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <font face=\"Times New Roman\" size=\"+2\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large e\"> is NOT the antilogarithm.  <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large e\"> is the base of the natural logs, that is: <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \ln(x)\ =\ \log_e(x)\">\r<BR>\n" );
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document.write( "If one half-life has elapsed, then <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\Large \frac{y}{n}\ =\ \frac{1}{2}\">.\r<BR>\n" );
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document.write( "Since the half-life of the substance in question is 32 years, then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ e^{32k}\ =\ \frac{1}{2} \">\r<BR>\n" );
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document.write( "Take the natural log of both sides:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \ln\left(e^{32k}\right)\ =\ \ln\left(\frac{1}{2}\right)\">\r<BR>\n" );
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document.write( "Use\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log_b(x^n)\ =\ n\log_b(x)\">\r<BR>\n" );
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document.write( "to write\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 32k\ln\left(e\right)\ =\ \ln\left(\frac{1}{2}\right)\">\r<BR>\n" );
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document.write( "Use \"the difference of the logs is the log of the quotient\",\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log_b(x) - \log_b(y) = \log_b\left(\frac{x}{y}\right)\">\r<BR>\n" );
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document.write( "to write\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 32k\ln\left(e\right)\ =\ \ln(1)\ -\ \ln(2)\"> \r<BR>\n" );
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document.write( "Use\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log_b(b)\ =\ 1\">\r<BR>\n" );
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document.write( "to write\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 32k\ =\ \ln(1)\ -\ \ln(2)\">\r<BR>\n" );
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document.write( "then use\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ \log_b(1)\ =\ 0\">\r<BR>\n" );
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document.write( "to write\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ 32k\ =\ -\ln(2)\">\r<BR>\n" );
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document.write( "then\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \ \ \ \ \ \ \ \ \ \ k =\ \frac{-\ln(2)}{32}\ \approx\ -0.022\">\r<BR>\n" );
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document.write( "John<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE e^{i\pi} + 1 = 0\"><BR>\n" );
document.write( "My calculator said it, I believe it, that settles it<BR>\n" );
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