document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=207881>Question 207881</A>This question is from textbook <A HREF=http://www.amazon.com/exec/obidos/ASIN/9780618825158/algebrahomeworkh>College Algebra and Trigonometry</A><BR>\n" );
document.write( ": <I><SPAN STYLE=\"word-break: break-all;\"> How do you solve a logarithmic word problem of this nature.<BR>\n" );
document.write( "LAW  A lawyer has determined that the number of people P(t)in a city of 1,200,000 people who have been exposed to a news item after t days is given by the function P(t)=1,200,000(1-e^-0.03t)\r<BR>\n" );
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document.write( "a. How many days after major crime has been reported have 40% of the population heard of the crime?<BR>\n" );
document.write( "b. A defense lawyer knows it will be very difficult to pick an unbiased jury after 80% of the population have heard of the crime. After how many days will 80% of the population have heard of the crime?    </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #157250 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=157250\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A lawyer has determined that the number of people P(t)in a city of 1,200,000 people who have been exposed to a news item after t days is given by the function P(t)=1,200,000(1-e^-0.03t)<BR>\n" );
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document.write( "PROBLEM A<BR>\n" );
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document.write( "a. How many days after major crime has been reported have 40% of the population heard of the crime?<BR>\n" );
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document.write( "p(t) = 1,200,000 * (1-e^(-.03*t)<BR>\n" );
document.write( "----<BR>\n" );
document.write( "when 40% of the population hear of the crime, this means that p(t) = 480,000<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "formula becomes:<BR>\n" );
document.write( "480,000 = 1,200,000 * (1-e^(-.03*t)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "remove the parentheses to get:<BR>\n" );
document.write( "480,000 = 1,200,000 - (1,200,000) * (e^(-.03*t))<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "subtract 1,200,000 from both sides of the equation to get:<BR>\n" );
document.write( "-720,000 = -(1,200,000) * (e^(-.03*t))<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "divide both sides of the equation by (-1,200,000) to get:<BR>\n" );
document.write( "(-720,000)/(-1,200,000) = e^(-.03*t))<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( ".6 = e^(-.03*t))<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the equation is now in a form that you can solve using logarithms.<BR>\n" );
document.write( "you can use LOG function of your calculator or LN function of your calculator.<BR>\n" );
document.write( "it doesn't matter which.<BR>\n" );
document.write( "you'll get the same either way.<BR>\n" );
document.write( "we'll use the LOG function of the calculator.<BR>\n" );
document.write( "----<BR>\n" );
document.write( "take the log of both sides of this equation to get:<BR>\n" );
document.write( "log(.6) = log(e^(-.03*t))<BR>\n" );
document.write( "by the laws of logarithms, this becomes:<BR>\n" );
document.write( "log(.6) = (-.03*t) * (log(e))<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "divide both sides of this equation by log(e) to get:<BR>\n" );
document.write( "log(.6)/log(e) = -.03*t<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "-.03*t = -.510825624<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since our exponent for e is -.03*t we'll leave the answer as is for now and just use that answer to verify that we have the right number for the exponent of e.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the equation we are trying to solve is:<BR>\n" );
document.write( "480,000 = 1,200,000 * (1-e^(-.03*t)<BR>\n" );
document.write( "substituting -.510825624 for -.03*t, that equation becomes:<BR>\n" );
document.write( "480,000 = 1,200,000 * (1 - e^(-.510825624))<BR>\n" );
document.write( "using the calculator to find e^(-.510825624) our equation becomes:<BR>\n" );
document.write( "480,000 = 1,200,000 * (1 - .6) which becomes:<BR>\n" );
document.write( "480,000 = 1,200,000 * (.4) which becomes:<BR>\n" );
document.write( "480,000 = 480,000 proving that the exponent we calculated was good.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we still haven't answered the question though.<BR>\n" );
document.write( "that question was how many days which is represented by t.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since we know that -.03*t = -.510825624 then we can solve for t by getting:<BR>\n" );
document.write( "t = (-.510825624) / (-.03) = 17.02752079<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "our answer is that 40% of the population will have read about the crime in 17.02752079 days which rounds down to 17 days.<BR>\n" );
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document.write( "PROBLEM B<BR>\n" );
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document.write( "b. A defense lawyer knows it will be very difficult to pick an unbiased jury after 80% of the population have heard of the crime. After how many days will 80% of the population have heard of the crime? <BR>\n" );
document.write( "-----<BR>\n" );
document.write( "problem b can be solved using the same method as in problem a.<BR>\n" );
document.write( "only the numbers will change.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "p(t) = 1,200,000 * (1-e^(-.03*t)<BR>\n" );
document.write( "----<BR>\n" );
document.write( "when 80% of the population hear of the crime, this means that p(t) = 600,000<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "formula becomes:<BR>\n" );
document.write( "960,000 = 1,200,000 * (1-e^(-.03*t)<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "remove the parentheses to get:<BR>\n" );
document.write( "960,000 = 1,200,000 - (1,200,000) * (e^(-.03*t))<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "subtract 1,200,000 from both sides of the equation to get:<BR>\n" );
document.write( "-240,000 = -(1,200,000) * (e^(-.03*t))<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "divide both sides of the equation by (-1,200,000) to get:<BR>\n" );
document.write( "(-720,000)/(-1,200,000) = e^(-.03*t))<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( ".2 = e^(-.03*t))<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the equation is now in a form that you can solve using logarithms.<BR>\n" );
document.write( "you can use LOG function of your calculator or the LN function of your calculator.<BR>\n" );
document.write( "it doesn't matter which.<BR>\n" );
document.write( "you'll get the same either way.<BR>\n" );
document.write( "we'll use the LN function of the calculator this time, just to be different.<BR>\n" );
document.write( "LN means natural logarithms which are taken to the base e.<BR>\n" );
document.write( "LOG means common logarithms which are taken to the base 10.<BR>\n" );
document.write( "----<BR>\n" );
document.write( "take the natural log of both sides of this equation to get:<BR>\n" );
document.write( "ln(.2) = ln(e^(-.03*t))<BR>\n" );
document.write( "by the laws of logarithms, this becomes:<BR>\n" );
document.write( "ln(.2) = (-.03*t) * (ln(e))<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "divide both sides of this equation by ln(e) to get:<BR>\n" );
document.write( "ln(.2)/ln(e) = -.03*t<BR>\n" );
document.write( "this becomes:<BR>\n" );
document.write( "-.03*t = -1.609437912<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since our exponent for e is -.03*t we'll leave the answer as is for now and just use that answer to verify that we have the right number for the exponent of e.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "the equation we are trying to solve is:<BR>\n" );
document.write( "960,000 = 1,200,000 * (1-e^(-.03*t)<BR>\n" );
document.write( "substituting -1.609437912 for -.03*t, that equation becomes:<BR>\n" );
document.write( "960,000 = 1,200,000 * (1 - e^(-1.609437912))<BR>\n" );
document.write( "using the calculator to find e^(-1.609437912) our equation becomes:<BR>\n" );
document.write( "960,000 = 1,200,000 * (1 - .2) which becomes:<BR>\n" );
document.write( "960,000 = 1,200,000 * (.8) which becomes:<BR>\n" );
document.write( "960,000 = 960,000 proving that the exponent we calculated was good.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "we still haven't answered the question though.<BR>\n" );
document.write( "that question was how many days which is represented by t.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "since we know that -.03*t = -1.609437912 then we can solve for t by getting:<BR>\n" );
document.write( "t = (-1.609437912) / (-.03) = 53.64793041<BR>\n" );
document.write( "-----<BR>\n" );
document.write( "our answer is that 80% of the population will have read about the crime in 53.64793041 days which rounds up to 54 days.<BR>\n" );
document.write( "-----<BR>\n" );
document.write( " </SPAN>\n" );
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