document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=92161>Question 92161</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Find the solution sets for the following:\r<BR>\n" );
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document.write( "In (3x) = 6 </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #67006 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=67006\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Given:<BR>\n" );
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document.write( "ln (3x) = 6<BR>\n" );
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document.write( "By the rules of logarithms, the log of a product is equal to the sum of the logs of each<BR>\n" );
document.write( "of the terms in the product.  Therefore, ln(3*x) = ln(3) + ln(x). Substitute this into the<BR>\n" );
document.write( "given equation and it becomes:<BR>\n" );
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document.write( "ln(3) + ln(x) = 6<BR>\n" );
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document.write( "Subtract ln(3) from both sides to get rid of that term on the left side. After this <BR>\n" );
document.write( "subtraction the equation is:<BR>\n" );
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document.write( "ln(x) = 6 - ln(3)<BR>\n" );
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document.write( "But ln(3) is just a number that you can get from a scientific calculator.  Enter 3 and then<BR>\n" );
document.write( "press the ln key. You will find that ln(3) is equal to 1.098612289. Replace ln(3) in <BR>\n" );
document.write( "the equation with that number and you get:<BR>\n" );
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document.write( "ln(x) = 6 - 1.098612289<BR>\n" );
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document.write( "Perform the subtraction on the right side of the equation and the equation reduces to:<BR>\n" );
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document.write( "ln(x) = 4.901387711<BR>\n" );
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document.write( "Next convert this logarithmic equation into it equivalent exponential form using the following<BR>\n" );
document.write( "rule of logarithms:<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;\" > is equivalent to <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( "And in this problem the base, b, of the natural logarithms is the number e (which is<BR>\n" );
document.write( "2.718281828).<BR>\n" );
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document.write( "So converting to exponential form results in an equation transformation of:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=ln%28x%29+=+4.90138771 ALIGN=MIDDLE  ALT=\"ln%28x%29+=+4.90138771\"   DISABLED_style= \"max-width:90%; height:auto;\" > converting to <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E4.90138771+=+x ALIGN=MIDDLE  ALT=\"e%5E4.90138771+=+x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and since e is approximately 2.718281828 this becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+2.718281828%5E4.90138771 ALIGN=MIDDLE  ALT=\"x+=+2.718281828%5E4.90138771\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Using a scientific calculator to raise 2.718281828 to the 4.90138771 power results in<BR>\n" );
document.write( "an answer of:<BR>\n" );
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document.write( "x = 134.4762643<BR>\n" );
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document.write( "You also could have done the problem by beginning with the conversion to exponential<BR>\n" );
document.write( "form directly. That is, starting with:<BR>\n" );
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document.write( "ln(3x) = 6<BR>\n" );
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document.write( "immediately raise the base e to the sixth power and set it equal to 3x as follows:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E6+=+3x ALIGN=MIDDLE  ALT=\"e%5E6+=+3x\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then divide both sides by 3 to find that:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+=+%28e%5E6%29%2F3 ALIGN=MIDDLE  ALT=\"x+=+%28e%5E6%29%2F3\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Then you can raise e to the sixth power <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2.718281828%5E6 ALIGN=MIDDLE  ALT=\"2.718281828%5E6\"   DISABLED_style= \"max-width:90%; height:auto;\" > and get an answer for this<BR>\n" );
document.write( "of 403.4287935 which you then divide by 3 to find that x = 134.4762645<BR>\n" );
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document.write( "The slight difference in the very last decimal place between this answer and the answer we<BR>\n" );
document.write( "got using the first method involves just calculator round off in all the steps of the<BR>\n" );
document.write( "first method. This second method is a little more accurate because the round off only occurs<BR>\n" );
document.write( "in the very last calculation.<BR>\n" );
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document.write( "Hope this helps you to understand the problem a little better.  The basis of answering<BR>\n" );
document.write( "this problem is knowing how to convert from logarithmic form to exponential form and you<BR>\n" );
document.write( "need to memorize that conversion. It gets used a lot in solving logarithmic equations.<BR>\n" );
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