document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=250805>Question 250805</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The population growth of an animal species is described by<BR>\n" );
document.write( "F(t) = 700 + 50 log3 (2t + 1),<BR>\n" );
document.write( "where t is measured in months. Find the population of this species in an area 13 month(s) after the species is introduced.\r<BR>\n" );
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document.write( "Thank you for any help you can offer. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #183337 by </B><A HREF=/tutors/aboutme.mpl?userid=jsmallt9><B>jsmallt9(3758)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jsmallt9><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=183337\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> So we have a value for t and we've been asked to find the value of F(t):<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%2813%29+=+700+%2B+50%2Alog%283%2C+%282%2813%29+%2B+1%29%29 ALIGN=MIDDLE  ALT=\"F%2813%29+=+700+%2B+50%2Alog%283%2C+%282%2813%29+%2B+1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Simplifying we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%2813%29+=+700+%2B+50%2Alog%283%2C+%2826+%2B+1%29%29 ALIGN=MIDDLE  ALT=\"F%2813%29+=+700+%2B+50%2Alog%283%2C+%2826+%2B+1%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%2813%29+=+700+%2B+50%2Alog%283%2C+%2827%29%29 ALIGN=MIDDLE  ALT=\"F%2813%29+=+700+%2B+50%2Alog%283%2C+%2827%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "At this point we want to find the base 3 logarithm of 27. I don't know of a calculator that can find base 3 logarithms. So what do we do?<ul><li>A logarithm can be found \"by hand\" if the argument is a power of the base of the logarithm.</li><li>We can use the base conversion formula for logarithms: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28q%29%29+=+log%28b%2C+%28q%29%29%2Flog%28b%2C+%28a%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28q%29%29+=+log%28b%2C+%28q%29%29%2Flog%28b%2C+%28a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > This gives us a way to convert a logarithm from one base, \"a\", into an expression of another base, \"b\".</li><li>If you can't remember the base conversion formula, like me, then you can derive it:<ul><li>Use a temporary variable. Let <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+log%28a%2C+%28q%29%29 ALIGN=MIDDLE  ALT=\"p+=+log%28a%2C+%28q%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li>Rewrite this in exponential form:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5Ep+=+q ALIGN=MIDDLE  ALT=\"a%5Ep+=+q\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li>Find the base b logarithm of both sides:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C+%28a%5Ep%29%29+=+log%28b%2C+%28q%29%29 ALIGN=MIDDLE  ALT=\"log%28b%2C+%28a%5Ep%29%29+=+log%28b%2C+%28q%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li>Use the property of logairhtms, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C+%28x%5Ey%29%29+=+y%2Alog%28b%2C+%28x%29%29 ALIGN=MIDDLE  ALT=\"log%28b%2C+%28x%5Ey%29%29+=+y%2Alog%28b%2C+%28x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to move the exponent in the argument out front:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p%2Alog%28b%2C+%28a%29%29+=+log%28b%2C+%28q%29%29 ALIGN=MIDDLE  ALT=\"p%2Alog%28b%2C+%28a%29%29+=+log%28b%2C+%28q%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li>Divide both sides by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C+%28a%29%29 ALIGN=MIDDLE  ALT=\"log%28b%2C+%28a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=p+=+log%28b%2C+%28q%29%29%2Flog%28b%2C+%28a%29%29 ALIGN=MIDDLE  ALT=\"p+=+log%28b%2C+%28q%29%29%2Flog%28b%2C+%28a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li><li>Replace the temporary variable with the expression it represents:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28q%29%29+=+log%28b%2C+%28q%29%29%2Flog%28b%2C+%28a%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28q%29%29+=+log%28b%2C+%28q%29%29%2Flog%28b%2C+%28a%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ></li></ul></li></ul><BR>\n" );
document.write( "With our problem the argument of our logarithm, 27, is a power of the base, 3. <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=27+=+3%5E3 ALIGN=MIDDLE  ALT=\"27+=+3%5E3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%283%2C+%2827%29%29+=+3 ALIGN=MIDDLE  ALT=\"log%283%2C+%2827%29%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >. If we do not recognize that 27 is a power of 3 then we must use the base conversion formula to convert the base into one our calculators can find:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%283%2C+%2827%29%29+=+log%28%2827%29%29%2Flog%28%283%29%29 ALIGN=MIDDLE  ALT=\"log%283%2C+%2827%29%29+=+log%28%2827%29%29%2Flog%28%283%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "or<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%283%2C+%2827%29%29+=+ln%2827%29%2Fln%283%29 ALIGN=MIDDLE  ALT=\"log%283%2C+%2827%29%29+=+ln%2827%29%2Fln%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Correctly using either of these with our calculator should give us 3 (or a decimal very, very close to 3).<br><BR>\n" );
document.write( "Now that we have figured out that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%283%2C+%2827%29%29+=+3 ALIGN=MIDDLE  ALT=\"log%283%2C+%2827%29%29+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, our equation becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%2813%29+=+700+%2B+50%2A3 ALIGN=MIDDLE  ALT=\"F%2813%29+=+700+%2B+50%2A3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "which simplifies to<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%2813%29+=+700+%2B+150 ALIGN=MIDDLE  ALT=\"F%2813%29+=+700+%2B+150\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=F%2813%29+=+850 ALIGN=MIDDLE  ALT=\"F%2813%29+=+850\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );