document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=84553>Question 84553</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello,<BR>\n" );
document.write( "I am having a problem with figuring this problem.  I have been trying to figure out how you come up with the answers and how to solve it.\r<BR>\n" );
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document.write( "Ln(12000 divided by 25000)<BR>\n" );
document.write( "This is underlined and underneath is<BR>\n" );
document.write( "0.075\r<BR>\n" );
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document.write( "Thanks for your help<BR>\n" );
document.write( "Michelle </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #60914 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=60914\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> If I understand your problem correctly, it is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ln%2812000%2F25000%29%29%2F0.075 ALIGN=MIDDLE  ALT=\"%28ln%2812000%2F25000%29%29%2F0.075\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If this is your problem, the first thing you can do is to cancel the zeros in the numerator<BR>\n" );
document.write( "with those of the denominator to get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ln%2812%2F25%29%29%2F0.075 ALIGN=MIDDLE  ALT=\"%28ln%2812%2F25%29%29%2F0.075\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can now recognize that you are taking the natural logarithm of a division.  By the rules<BR>\n" );
document.write( "of logarithms you can write this as the difference of two logarithms ... the logarithm<BR>\n" );
document.write( "of the numerator minus the logarithm of the denominator.  So your problem is equivalent to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28ln%2812%29-ln%2825%29%29%2F0.075 ALIGN=MIDDLE  ALT=\"%28ln%2812%29-ln%2825%29%29%2F0.075\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You can now use a scientific calculator to find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%2812%29+=+2.48490665 ALIGN=MIDDLE  ALT=\"ln%2812%29+=+2.48490665\"   DISABLED_style= \"max-width:90%; height:auto;\" > and that<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%2825%29+=+3.218875825 ALIGN=MIDDLE  ALT=\"ln%2825%29+=+3.218875825\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substituting these two values into the problem makes it become:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282.48490665+-+3.218875825%29%2F0.075 ALIGN=MIDDLE  ALT=\"%282.48490665+-+3.218875825%29%2F0.075\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Performing the subtraction in the numerator results in the problem becoming:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-0.733969175%2F0.075 ALIGN=MIDDLE  ALT=\"-0.733969175%2F0.075\"   DISABLED_style= \"max-width:90%; height:auto;\" > <BR>\n" );
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document.write( "and, finally, doing the division results in an answer of:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9.786255668 ALIGN=MIDDLE  ALT=\"-9.786255668\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "And since we used a scientific calculator to find ln(12) and ln(25) we could have done <BR>\n" );
document.write( "the whole thing on a scientific calculator from the very start. Basically, it could have<BR>\n" );
document.write( "been done on a typical calculator as follows:<BR>\n" );
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document.write( "Enter 12000<BR>\n" );
document.write( "Press the divide key<BR>\n" );
document.write( "Enter 25000<BR>\n" );
document.write( "Press the \"=\" key<BR>\n" );
document.write( "At this point you have found 12000 divided by 25000<BR>\n" );
document.write( "Press the natural logarithm \"ln\" key<BR>\n" );
document.write( "The above step gives you the natural logarithm of 12000 divided by 25000<BR>\n" );
document.write( "Press the divide key<BR>\n" );
document.write( "Enter 0.075<BR>\n" );
document.write( "Press the \"=\" key<BR>\n" );
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document.write( "You should have the answer of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-9.786255668 ALIGN=MIDDLE  ALT=\"-9.786255668\"   DISABLED_style= \"max-width:90%; height:auto;\" > which agrees with what we got by going<BR>\n" );
document.write( "through a few more steps. But I suppose it was OK to do the extra steps just for practice.<BR>\n" );
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document.write( "Hope this helps you to understand logarithms in general a little better, and how to work<BR>\n" );
document.write( "with natural logarithms in particular.<BR>\n" );
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document.write( "If I mis-interpreted what you meant your original problem to be, please repost the entire<BR>\n" );
document.write( "problem. </SPAN>\n" );
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