document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=692322>Question 692322</A>: <I><SPAN STYLE=\"word-break: break-all;\"> solve the following equations (give the exact answer) and then approximate the answer by rounding to 3 decimal places.\r<BR>\n" );
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document.write( "2/5=4/1+e^6x </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #427103 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=427103\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> What you posted means<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F5=4%2F1%2Be%5E6x ALIGN=MIDDLE  ALT=\"2%2F5=4%2F1%2Be%5E6x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "But for various reasons I suspect that what you meant was:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%2F5=4%2F%281%2Be%5E%286x%29%29 ALIGN=MIDDLE  ALT=\"2%2F5=4%2F%281%2Be%5E%286x%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "If I am correct, then please use parentheses to clarify your expressions. For example:<BR>\n" );
document.write( "2/5=4/(1+e^(6x))<BR>\n" );
document.write( "If I am wrong about your equation then the rest of this will not help you.<br><BR>\n" );
document.write( "First let's eliminate the fractions by multiplying both sides by the lowest common denominator, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%281%2Be%5E%286x%29%29 ALIGN=MIDDLE  ALT=\"5%281%2Be%5E%286x%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=5%281%2Be%5E%286x%29%29%282%2F5%29=5%281%2Be%5E%286x%29%29%284%2F%281%2Be%5E%286x%29%29%29 ALIGN=MIDDLE  ALT=\"5%281%2Be%5E%286x%29%29%282%2F5%29=5%281%2Be%5E%286x%29%29%284%2F%281%2Be%5E%286x%29%29%29\"   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=2%2B2e%5E%286x%29=20 ALIGN=MIDDLE  ALT=\"2%2B2e%5E%286x%29=20\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Next let's isolate the base and its exponent. Subtracting 2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2e%5E%286x%29=18 ALIGN=MIDDLE  ALT=\"2e%5E%286x%29=18\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Divide by 2:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E%286x%29=9 ALIGN=MIDDLE  ALT=\"e%5E%286x%29=9\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><BR>\n" );
document.write( "Now we use logarithms. Any base of logarithms may be used. But there are advantages to choosing certain bases:<ul><li>Choosing a base of logarithm that matches the base of the exponent will result in the simplest possible expression for the solution.</li><li>Choosing a base of logarithm that your calculator \"knows\" (base 10, \"log\", or base e, \"ln\") will result in an expression that will be easy to convert into a decimal approximation.</li></ul>In this equation, with the base of the exponent being e, we can get both advantages by using ln:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28e%5E%286x%29%29=ln%289%29 ALIGN=MIDDLE  ALT=\"ln%28e%5E%286x%29%29=ln%289%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next we use a property of logarithms, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%5Eq%29%29+=+q%2Alog%28a%2C+%28p%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, which let's us move the exponent of the argument out in front. (It is this very property that is the reason we use logarithms on equations like this. It give us a way to move the exponent, where the variable is, out in front where we can then \"get at\" the variable with \"regular\" algebra.) Using this property we get:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%286x%29ln%28e%29=ln%289%29 ALIGN=MIDDLE  ALT=\"%286x%29ln%28e%29=ln%289%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "By definition ln(e) = 1. (This is why matching the bases of the exponent and logarithm leads to the simplest possible expression for the solution.)<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=6x=ln%289%29 ALIGN=MIDDLE  ALT=\"6x=ln%289%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Now we can solve for x. Dividing by 6:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=ln%289%29%2F6 ALIGN=MIDDLE  ALT=\"x=ln%289%29%2F6\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "This is an exact expression for the solution to your equation. You were also asked to find a rounded decimal approximation for the answer. I'll leave that up to you and your calculator. (Just make sure you find ln(9) first and then divide by 6.)<BR>\n" );
document.write( " </SPAN>\n" );
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