document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=886286>Question 886286</A>: <I><SPAN STYLE=\"word-break: break-all;\"> 2^(log_2(3)-log_8(9))=<BR>\n" );
document.write( "so far i used the change of base to get<BR>\n" );
document.write( "2^(ln3/ln8)=<BR>\n" );
document.write( "the answer is 3root3 ->^3&#8730;3 but i don't know how to get that from here. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #535888 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><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/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=535888\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> i didn't get 3 * sqrt(3) as the value for 2^(log2(3) - log8(9))<BR>\n" );
document.write( "my first look was to use the calculator and convert the logs to base 10 which the calculator can handle.\r<BR>\n" );
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document.write( "i got:<BR>\n" );
document.write( "log2(3) = log10(3)/log10(2) = 1.584962501<BR>\n" );
document.write( "log8(9) = log10(9)/log10(8) = 1.056641667\r<BR>\n" );
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document.write( "putting these into the original equation and i got:<BR>\n" );
document.write( "2^(log2(3)-log8(9)) = 2^(1.584962501 - 1.056641667) = 2^(.5283208336) which is equal to 1.44224957\r<BR>\n" );
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document.write( "so your solution should be 1.44224957\r<BR>\n" );
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document.write( "now to solve it the long way that takes a lot more work but may also be instructive.\r<BR>\n" );
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document.write( "log8(9) = y if and only if 8^y = 9<BR>\n" );
document.write( "since 8 is equal to 2^3, this equation becomes:<BR>\n" );
document.write( "(2^3)^y = 9 which is equivalent to 2^3y = 9<BR>\n" );
document.write( "2^3y = 9 if and only if log2(9) = 3y<BR>\n" );
document.write( "divide both sides of this equation by 3 to get y = log2(9)/3\r<BR>\n" );
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document.write( "you have both log8(9) and log2(9)/3 equal to y so these 2 expressions are also equal to each other.\r<BR>\n" );
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document.write( "replace log8(9) by log2(9)/3 in your original equation and you get:\r<BR>\n" );
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document.write( "2^(log2(3) - log2(9)/3)\r<BR>\n" );
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document.write( "place log2(3) - log2(9)/3 under the same denominator and you get:<BR>\n" );
document.write( "2^((3*log2(3) - log2(9)) / 3)\r<BR>\n" );
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document.write( "since a*log(b) = log(b^a), you can simplify this to get:<BR>\n" );
document.write( "2^(log2(3^3) - log2(9)) / 3)\r<BR>\n" );
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document.write( "simplify this further to get:<BR>\n" );
document.write( "2^((log2(27) - log2(9)) / 3)\r<BR>\n" );
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document.write( "since log(a) - log(b) = log(a/b), this expression becomes:<BR>\n" );
document.write( "2^(log2(27/9)/3)\r<BR>\n" );
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document.write( "simplify this further to get:<BR>\n" );
document.write( "2^(log2(3)/3)\r<BR>\n" );
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document.write( "once you get to this point, you still have to do some conversions.<BR>\n" );
document.write( "suffice it to say that 2^(log2(3)/3) = 1.44224957, same as we got originally.\r<BR>\n" );
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document.write( "if you use your calculator and take the cube root of 3, you will see that cube root of 3 is equal to 1.44224957.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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