document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=379858>Question 379858</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Having trouble with a logarithm!  I am okay with logarithms, but this one throws in a square root so I am lost as to what to do with it.\r<BR>\n" );
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document.write( "log base-2 (log base-b (sqrt b)) , where b > 0.\r<BR>\n" );
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document.write( "I just had a thought.. do I need to turn sqrt b into b^(1/2)?\r<BR>\n" );
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document.write( "Any help would be appreciated, thanks! :) </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #269695 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=269695\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C+%28log%28b%2C+%28sqrt%28b%29%29%29%29%29 ALIGN=MIDDLE  ALT=\"log%282%2C+%28log%28b%2C+%28sqrt%28b%29%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "You are correct. start by rewriting <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28b%29 ALIGN=MIDDLE  ALT=\"sqrt%28b%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > as <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5E%281%2F2%29 ALIGN=MIDDLE  ALT=\"b%5E%281%2F2%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C+%28log%28b%2C+%28b%5E%281%2F2%29%29%29%29%29 ALIGN=MIDDLE  ALT=\"log%282%2C+%28log%28b%2C+%28b%5E%281%2F2%29%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "Next, on the inner logarithm, we can 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;\" >, to \"move\" the exponent of the argument out in front of the logarithm:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C+%28%281%2F2%29%2Alog%28b%2C+%28b%29%29%29%29 ALIGN=MIDDLE  ALT=\"log%282%2C+%28%281%2F2%29%2Alog%28b%2C+%28b%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "By definition, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C+%28b%29%29+=+1 ALIGN=MIDDLE  ALT=\"log%28b%2C+%28b%29%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > so this becomes:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C+%281%2F2%29%29 ALIGN=MIDDLE  ALT=\"log%282%2C+%281%2F2%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The quick way to finish requires that we understand what this expression represents. It represents the exponent for 2 that results in 1/2. If you also understand exponents, you will know that the answer is: -1!<br><BR>\n" );
document.write( "The long way is to use another property of logarithms, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29 ALIGN=MIDDLE  ALT=\"log%28a%2C+%28p%2Fq%29%29+=+log%28a%2C+%28p%29%29+-+log%28a%2C+%28q%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, to split the numerator and denominator into separate logarithms:<BR>\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C+%281%29%29+-+log%282%2C+%282%29%29 ALIGN=MIDDLE  ALT=\"log%282%2C+%281%29%29+-+log%282%2C+%282%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><BR>\n" );
document.write( "The logarithm of 1, regardless of the base, is zero (because any number to the zero power is 1). And by definition <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%282%2C+%282%29%29+=+1 ALIGN=MIDDLE  ALT=\"log%282%2C+%282%29%29+=+1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So now we have:<BR>\n" );
document.write( "0 - 1<BR>\n" );
document.write( "which equals<BR>\n" );
document.write( "-1 </SPAN>\n" );
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