document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=117028>Question 117028</A>: <I><SPAN STYLE=\"word-break: break-all;\"> suppose that the function P=12+50ln  x represents the percentage of inbound e-mail in the U.S> that is considered spam, where x is the number of years after 2001.\r<BR>\n" );
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document.write( "carry all calculationjs to six decimals on each intermediate step, when necessary.\r<BR>\n" );
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document.write( "use this model to approximate the percentage of spam in the year 2005 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #85149 by </B><A HREF=/tutors/aboutme.mpl?userid=bucky><B>bucky(2189)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=bucky><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=85149\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given the equation:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=P=12%2B50%2Alnx ALIGN=MIDDLE  ALT=\"P=12%2B50%2Alnx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "In which P represents the percent (as a whole number) of the inbound e-mail in the U.S. that is spam,<BR>\n" );
document.write( "and x represents the number of years after 2001 that will determine what P is.<BR>\n" );
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document.write( "For this problem you are asked to solve for x if P is 90 percent. Begin by substituting<BR>\n" );
document.write( "90 for P in the equation to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=90=12%2B50%2Alnx ALIGN=MIDDLE  ALT=\"90=12%2B50%2Alnx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Get rid of the 12 on the right side by subtracting 12 from both sides. When you do that subtraction,<BR>\n" );
document.write( "you reduce the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=78=50%2Alnx ALIGN=MIDDLE  ALT=\"78=50%2Alnx\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Just to get this into a little more standard form ... with the unknown on the left side ...<BR>\n" );
document.write( "transpose (switch sides) this equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=50%2Alnx+=+78 ALIGN=MIDDLE  ALT=\"50%2Alnx+=+78\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Divide both sides by 50 to reduce the equation to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lnx+=+78%2F50+=+1.56 ALIGN=MIDDLE  ALT=\"lnx+=+78%2F50+=+1.56\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "which (leaving out the the step that shows 78 divided by 50} is just:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lnx+=+1.56 ALIGN=MIDDLE  ALT=\"lnx+=+1.56\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "This is a logarithmic form of an equation. It can be converted to an equivalent exponential form<BR>\n" );
document.write( "by applying a rule that says the logarithmic form:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2Cx%29+=+y ALIGN=MIDDLE  ALT=\"log%28b%2Cx%29+=+y\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "is equivalent to the exponential form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Ey+=+x ALIGN=MIDDLE  ALT=\"b%5Ey+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "In this problem b, the base of the logarithm, is e which is the base of the natural logarithms<BR>\n" );
document.write( "that are commonly indicated by \"ln\". By comparing the logarithmic form of this rule to the logarithmic<BR>\n" );
document.write( "form derived in our problem, you can see that b = e, y = 1.56, and x = x. Substitute these<BR>\n" );
document.write( "values into the exponential form of the rule and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E1.56+=+x ALIGN=MIDDLE  ALT=\"e%5E1.56+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "as being the exponential form that is equivalent to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=lnx+=+1.56 ALIGN=MIDDLE  ALT=\"lnx+=+1.56\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can use the exponential form to solve for x. Transpose the exponential form to get the<BR>\n" );
document.write( "x in the more standard arrangement of having the unknown on the left side and the exponential<BR>\n" );
document.write( "form becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+e%5E1.56 ALIGN=MIDDLE  ALT=\"x+=+e%5E1.56\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can now use a scientific calculator to solve for x. Just think of e as being a number that<BR>\n" );
document.write( "you are going to raise to the 1.56 power. [The value of e is approximately 2.718281828]<BR>\n" );
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document.write( "So you can use your calculator to raise 2.718281828 to the 1.56 power. Or look on your calculator<BR>\n" );
document.write( "for the function key that shows <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If you have one, your calculator might work <BR>\n" );
document.write( "this way: (1) enter 1.56, then (2) activate the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > key.<BR>\n" );
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document.write( "Either way you should get an answer of 4.758821245 which rounds off to 4.76. This tells you<BR>\n" );
document.write( "that 4.76 years after 2001 the inbound spam in the U.S. can be expected to be 90 percent of<BR>\n" );
document.write( "the message traffic.<BR>\n" );
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document.write( "(Assuming that this equation begins on December 31, 2001 ... 4.76 years later<BR>\n" );
document.write( "(4 and three-quarters years later) the spam level would reach 90 percent around September 30, 2006<BR>\n" );
document.write( "which is nearly 5 years after December 31, 2001 ... actually 4.75 years after December 31, 2001 <BR>\n" );
document.write( "... <BR>\n" );
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document.write( "Hope this helps you to see your way through this problem and helps you also to learn a little <BR>\n" );
document.write( "about working with logarithms ... especially natural logarithms (indicated by ln). </SPAN>\n" );
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