document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1107750>Question 1107750</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Determine all the values of x for which:\r<BR>\n" );
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document.write( "&#8730;x^(log10x)=100\r<BR>\n" );
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document.write( "(To be read as \"the square root of x to the power of log to the base 10 of x, equals 100\")\r<BR>\n" );
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document.write( "I tried to solve this, but I keep getting different answers. I think x=100, but how do I find the other possible values of x?\r<BR>\n" );
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document.write( "Thank you in advance. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #722783 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=722783\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> your equation is sqrt(x ^ log(x)) = 100\r<BR>\n" );
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document.write( "base 10 is implied if no base is shown.\r<BR>\n" );
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document.write( "swuare both sides of this equation to get:\r<BR>\n" );
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document.write( "x ^ log(x) = 10,000.\r<BR>\n" );
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document.write( "take the log of both sides of this equation to get:\r<BR>\n" );
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document.write( "log(x ^ log(x)) = log(10,000).\r<BR>\n" );
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document.write( "log(x ^ log(x) is equal to log(x) * log(x) which is equal to log(x)^2.\r<BR>\n" );
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document.write( "you get log(x)^2 = log(10,000).\r<BR>\n" );
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document.write( "log(10,000) is equal to 4.\r<BR>\n" );
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document.write( "therefore log(x)^2 = 4.\r<BR>\n" );
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document.write( "take the square root of both sides of this equation to get log(x) = plus or minus 2.\r<BR>\n" );
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document.write( "log(x) = 2 if and only if 10^2 = x.\r<BR>\n" );
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document.write( "that makes x = 100.\r<BR>\n" );
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document.write( "log(x) = -2 if and only if 10^-2 = x.\r<BR>\n" );
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document.write( "10^-2 is equal to 1/100.\r<BR>\n" );
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document.write( "therefore 1/100 = x.\r<BR>\n" );
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document.write( "looks like x can either be 100 or 1/100.\r<BR>\n" );
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document.write( "replace x with each of these values to see if the original equation holds true.\r<BR>\n" );
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document.write( "the original equation is sqrt(x ^ log(x)) = 100\r<BR>\n" );
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document.write( "when x = 100, this becomes sqrt(100 ^ log(100)) = 100.\r<BR>\n" );
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document.write( "since log(100) = 2, this equation becomes sqrt(100 ^ 2) = 100.\r<BR>\n" );
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document.write( "since sqrt(100^2) = 100, you get 100 = 100, confirming that x = 100 is a solution to the original equation.\r<BR>\n" );
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document.write( "now let x = 1/100.\r<BR>\n" );
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document.write( "the original equation is sqrt(x ^ log(x)) = 100\r<BR>\n" );
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document.write( "when x = 1/100, the original equation becomes sqrt((1/100) ^ log(1/100)) = 100.\r<BR>\n" );
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document.write( "log(1/100) = -2, therefore you get sqrt((1/100) ^ -2) = 100.\r<BR>\n" );
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document.write( "(1/100) ^ -2 is equal to 10,000.\r<BR>\n" );
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document.write( "(1/100)^ -2 = 10,000.\r<BR>\n" );
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document.write( "the equation becomes sqrt(10,000) = 100.\r<BR>\n" );
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document.write( "this results in 100 = 100 which is true, confirming that x can either be 100 or 1/100.\r<BR>\n" );
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document.write( "that's your solution.\r<BR>\n" );
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document.write( " </SPAN>\n" );
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