document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=124442>Question 124442</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Use the change of base formula to solve log10 (e) \r<BR>\n" );
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document.write( "A 0.891<BR>\n" );
document.write( "B 1.458<BR>\n" );
document.write( "C 0.434<BR>\n" );
document.write( "D 1.653<BR>\n" );
document.write( "E 0.422<BR>\n" );
document.write( "F 0.123\r<BR>\n" );
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document.write( "I tried what I thought was the correct way of solving this,but i came to an answer which isnt a choice,so im not sure what to try.<BR>\n" );
document.write( "If someone could show me the steps to come to the correct answer it would be very much appriciated. Thank you very much for your time!<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #91255 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=91255\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Just for grins, let's find the answer using a calculator. First let's find the value of e<BR>\n" );
document.write( "by entering 1 on the calculator, then pressing the <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5Ex ALIGN=MIDDLE  ALT=\"e%5Ex\"   DISABLED_style= \"max-width:90%; height:auto;\" > key. You should see the answer <BR>\n" );
document.write( "2.718281828 appear. This tells you that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=e%5E1 ALIGN=MIDDLE  ALT=\"e%5E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > which is just e is equal to 2.718281828.<BR>\n" );
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document.write( "Then to find the log to the base 10 of that value, just press the \"log\" key. The answer that<BR>\n" );
document.write( "should appear is 0.434294481 which rounds to 0.434. Now we know the answer to the problem.<BR>\n" );
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document.write( "Now let's see if we can get that answer by using the change of base formula.<BR>\n" );
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document.write( "The change of base formula is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2Cx%29+=+%28log%28b%2Cx%29%29%2F%28log%28b%2Ca%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2Cx%29+=+%28log%28b%2Cx%29%29%2F%28log%28b%2Ca%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "You are asked to use that formula to find:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Ce%29 ALIGN=MIDDLE  ALT=\"log%2810%2Ce%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "By comparing the given problem to the left side of the change of base formula you can<BR>\n" );
document.write( "see that a = 10 and x = e. Here comes the \"thinking part.\" The natural log of e is 1. This means:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28e%2Ce%29+=+1 ALIGN=MIDDLE  ALT=\"log%28e%2Ce%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "So let's say that b = e. Once we say that, we can substitute a = 10, x = e, and b = e into the <BR>\n" );
document.write( "change of base formula. Start with:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2Cx%29+=+%28log%28b%2Cx%29%29%2F%28log%28b%2Ca%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2Cx%29+=+%28log%28b%2Cx%29%29%2F%28log%28b%2Ca%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Make the substitutions and you have:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Ce%29+=+%28log%28e%2Ce%29%29%2F%28log%28e%2C10%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2Ce%29+=+%28log%28e%2Ce%29%29%2F%28log%28e%2C10%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "but as we already know, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28e%2Ce%29+=+1 ALIGN=MIDDLE  ALT=\"log%28e%2Ce%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So we can substitute 1 for that and the change<BR>\n" );
document.write( "of base formula you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Ce%29+=+1%2F%28log%28e%2C10%29%29 ALIGN=MIDDLE  ALT=\"log%2810%2Ce%29+=+1%2F%28log%28e%2C10%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Use your calculator to calculate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28e%2C10%29 ALIGN=MIDDLE  ALT=\"log%28e%2C10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. On your calculator, enter 10 and press the \"ln\" <BR>\n" );
document.write( "key to find that it equals 2.302585093. Substitute that value for <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28e%2C10%29 ALIGN=MIDDLE  ALT=\"log%28e%2C10%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Ce%29+=+%281%2F2.302585093 ALIGN=MIDDLE  ALT=\"log%2810%2Ce%29+=+%281%2F2.302585093\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Do the division on the right side and you get:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2Ce%29+=+0.434294481 ALIGN=MIDDLE  ALT=\"log%2810%2Ce%29+=+0.434294481\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "and this rounds off to:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C3%29+=+0.434 ALIGN=MIDDLE  ALT=\"log%2810%2C3%29+=+0.434\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Unfortunately, this doesn't show why the change of base formula is really useful. As<BR>\n" );
document.write( "long as you are working in the common bases 10 and e, you can pretty much do the math on a<BR>\n" );
document.write( "calculator because those functions are on a calculator or can be evaluated from tables commonly<BR>\n" );
document.write( "found in math books.<BR>\n" );
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document.write( "Where the change of base formula becomes really important is when you have to work with <BR>\n" );
document.write( "logarithms in bases other that 10 or e. To find logarithms in unusual bases, you use the<BR>\n" );
document.write( "change of base formula to convert from logs in common bases (10 and e) to get logs in<BR>\n" );
document.write( "other bases. It's a powerful tool.<BR>\n" );
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document.write( "For example, suppose you need to find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%286%2C50%29 ALIGN=MIDDLE  ALT=\"log%286%2C50%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. You don't find that in any common math<BR>\n" );
document.write( "book or on a common scientific calculator. You can find <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C50%29 ALIGN=MIDDLE  ALT=\"log%2810%2C50%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So start with<BR>\n" );
document.write( "the change of base formula.<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2Cx%29+=+%28log%28b%2Cx%29%29%2F%28log%28b%2Ca%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2Cx%29+=+%28log%28b%2Cx%29%29%2F%28log%28b%2Ca%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "If you substitute 6 for a and 50 for x this formula becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%286%2C50%29+=+log%28b%2C50%29%2Flog%28b%2C6%29 ALIGN=MIDDLE  ALT=\"log%286%2C50%29+=+log%28b%2C50%29%2Flog%28b%2C6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now all you have to do is to select for b some base that you can work in easily. For example,<BR>\n" );
document.write( "let b = 10. Substituting this into the formula that is being derived results in:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%286%2C50%29+=+log%2810%2C50%29%2Flog%2810%2C6%29 ALIGN=MIDDLE  ALT=\"log%286%2C50%29+=+log%2810%2C50%29%2Flog%2810%2C6%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Now use a calculator to find that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C50%29=+1.698970004 ALIGN=MIDDLE  ALT=\"log%2810%2C50%29=+1.698970004\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C6%29+=+0.77815125 ALIGN=MIDDLE  ALT=\"log%2810%2C6%29+=+0.77815125\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Substitute those values into the formula and it becomes:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%286%2C50%29+=+log%2810%2C50%29%2Flog%2810%2C6%29+=+1.698970004%2F0.77815125 ALIGN=MIDDLE  ALT=\"log%286%2C50%29+=+log%2810%2C50%29%2Flog%2810%2C6%29+=+1.698970004%2F0.77815125\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "Do the division and the result is:<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%286%2C50%29+=+2.183341611 ALIGN=MIDDLE  ALT=\"log%286%2C50%29+=+2.183341611\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
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document.write( "And that is typical of using the change of base formula to find logs to other bases.<BR>\n" );
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document.write( "Hope this helps you to understand the change of base formula.<BR>\n" );
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