document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1003768>Question 1003768</A>: <I><SPAN STYLE=\"word-break: break-all;\"> What is X in (3x)^(1+ log of X base 3)=3 and how do I get to it?  </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #620424 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=620424\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> as far as i can tell, x = 1\r<BR>\n" );
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document.write( "when x = 1, the equation of:\r<BR>\n" );
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document.write( "(3x)^(1 + log3(x)) = 3 becomes:\r<BR>\n" );
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document.write( "(3*1)^(1 + log3(1)) = 3 which becomes:\r<BR>\n" );
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document.write( "(3)^(1 + log3(1)) = 3\r<BR>\n" );
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document.write( "log3(1) = y if and only if 3^y = 1<BR>\n" );
document.write( "this can only occur if y = 0 because 3^0 = 1\r<BR>\n" );
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document.write( "therefore 1 + log3(1)) is equal to 1 + 0 which is equal to 1.\r<BR>\n" );
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document.write( "the equation becomes:\r<BR>\n" );
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document.write( "(3)^1 = 3 which becomes 3 = 3\r<BR>\n" );
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document.write( "the equation is true when x = 1.\r<BR>\n" );
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document.write( "how was this derived?\r<BR>\n" );
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document.write( "by magic of course.\r<BR>\n" );
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document.write( "here's the magic, if i did it correctly.\r<BR>\n" );
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document.write( "your equation is:\r<BR>\n" );
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document.write( "(3x)^(1+log3(x)) = 3\r<BR>\n" );
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document.write( "let y = 1 + log3(x) and your equation becomes:\r<BR>\n" );
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document.write( "(3x)^y = 3\r<BR>\n" );
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document.write( "this is true if and only if log3x(3) = y\r<BR>\n" );
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document.write( "log3x(3) is equal to log(3) / log(3x)\r<BR>\n" );
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document.write( "equation becomes log(3) / log(3x) = y\r<BR>\n" );
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document.write( "we had already set y = 1 + log3(x) \r<BR>\n" );
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document.write( "log3(x) is equal to log(x)/log(3)\r<BR>\n" );
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document.write( "therefore:\r<BR>\n" );
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document.write( "y = 1 + log3(x) becomes:\r<BR>\n" );
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document.write( "y = 1 + log(x)/log(3)\r<BR>\n" );
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document.write( "now we have:\r<BR>\n" );
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document.write( "y = 1 + log(x) / log(3) and we have:\r<BR>\n" );
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document.write( "y = log(3) / log(3x)\r<BR>\n" );
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document.write( "y = 1 + log(x) / log(3) is equivalent to y = (log(3) + log(x)) / log(3)\r<BR>\n" );
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document.write( "y = log(3) / log(3x) is equivalent to y = log(3) / (log(3) + log(x))\r<BR>\n" );
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document.write( "replace y in the second equation with y = (log(3) + log(x)) / log(3) from the first equation to get:\r<BR>\n" );
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document.write( "(log(3) + log(x)) / log(3) = log(3) / (log(3) + log(x)).\r<BR>\n" );
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document.write( "multiply both sides of this equation by log(3) and multiply both sides of this equation by (log(3) + log(x)) to get:\r<BR>\n" );
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document.write( "(log(3) + log(x))^2 = log(3)^2\r<BR>\n" );
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document.write( "take the square root of both sides of this equation to get:\r<BR>\n" );
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document.write( "log(3) + log(x) = log(3)\r<BR>\n" );
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document.write( "subtract log(3) from both sides of this equation to get:\r<BR>\n" );
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document.write( "log(x) = 0\r<BR>\n" );
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document.write( "solve for x to get x = 1.\r<BR>\n" );
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document.write( "there may be another, easier way to solve this, but i couldn't figure it out.\r<BR>\n" );
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document.write( "this was quite convoluted and i'm happy i was able to get an answer that made sense the way that i did do it.\r<BR>\n" );
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document.write( "hopefully you can understand what i did and why i did it.\r<BR>\n" );
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document.write( "some concepts involved.\r<BR>\n" );
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document.write( "y = logb(x) if and only if b^y = x\r<BR>\n" );
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document.write( "b^y = x if and only if logb(x) = y\r<BR>\n" );
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document.write( "log(a*b) = log(a) + log(b)\r<BR>\n" );
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document.write( "log(a^b) = b*log(a)\r<BR>\n" );
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document.write( "logb(x) = loga(x) / loga(b)\r<BR>\n" );
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document.write( "a + b/c = (ac + b) / c\r<BR>\n" );
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document.write( " </SPAN>\n" );
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