document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=606592>Question 606592</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello iam currently taking a college algebra class and my professor gave me this bonus problem y=log9(log 8(log 7(log 6(log 5(x)))))) and find the domain of dy/dx. i have no idea where or how to begin this problem any help is appreciated. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #382697 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><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=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=382697\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE y = \log_9 {\log_8 {\log_7 {\log_6{ \log_5{x}}}}}\">\r<BR>\n" );
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document.write( "First, we should find where x is defined. Note that the domain of any log function is always the positive real numbers (unless we are dealing with complex logarithms -- which we are not).\r<BR>\n" );
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document.write( "Hence,\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \log_8 {\log_7 {\log_6{ \log_5{x}}}} > 0\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \log_7 {\log_6{ \log_5{x}}} > 8^0 = 1\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \log_6{ \log_5{x}} > 7^{8^0} = 7\"> (see the pattern?)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE \log_5{x} > 6^{7^{8^0}} = 6^7\">\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x > 5^{ 6^{7^{8^0}}} = 5^{6^7}\">\r<BR>\n" );
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document.write( "Therefore any value of x such that <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE x > 5^{6^7}\"> is in the domain. Since log x is a differentiable function, log(log(x)) is differentiable, and so on. The domain of dy/dx is the same as the domain of y. </SPAN>\n" );
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