document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=198808>Question 198808</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Hello!\r<BR>\n" );
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document.write( "A function f(x) is given.<BR>\n" );
document.write( "f(x) = log5(log10(x))\r<BR>\n" );
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document.write( "(a) Find the domain of the function f\r<BR>\n" );
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document.write( "(b) Find the inverse function of f.\r<BR>\n" );
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document.write( "f -1(x) =\r<BR>\n" );
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document.write( "thanks for your help!\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #149307 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=149307\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> a)\r<BR>\n" );
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document.write( "Since you cannot take the log of 0 or a negative value, this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%29%29%3E0 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E0 ALIGN=MIDDLE  ALT=\"x%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > (for the outer and inner logs)\r<BR>\n" );
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document.write( "If we solve <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%2810%2C%28x%29%29%3E0 ALIGN=MIDDLE  ALT=\"log%2810%2C%28x%29%29%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E1 ALIGN=MIDDLE  ALT=\"x%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So combining the inequalities <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E0 ALIGN=MIDDLE  ALT=\"x%3E0\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E1 ALIGN=MIDDLE  ALT=\"x%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" > (ie perform a set union), we get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E1 ALIGN=MIDDLE  ALT=\"x%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >.\r<BR>\n" );
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document.write( "So the domain of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=log%285%2C%28log%2810%2C%28x%29%29%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=log%285%2C%28log%2810%2C%28x%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x%3E1 ALIGN=MIDDLE  ALT=\"x%3E1\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "------------------\r<BR>\n" );
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document.write( "b)\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=f%28x%29=log%285%2C%28log%2810%2C%28x%29%29%29%29 ALIGN=MIDDLE  ALT=\"f%28x%29=log%285%2C%28log%2810%2C%28x%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Start with the given function.\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=log%285%2C%28log%2810%2C%28x%29%29%29%29 ALIGN=MIDDLE  ALT=\"y=log%285%2C%28log%2810%2C%28x%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Replace f(x) with \"y\"\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=log%285%2C%28log%2810%2C%28y%29%29%29%29 ALIGN=MIDDLE  ALT=\"x=log%285%2C%28log%2810%2C%28y%29%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Switch each x and y\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5%5Ex=log%2810%2C%28y%29%29 ALIGN=MIDDLE  ALT=\"5%5Ex=log%2810%2C%28y%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite the equation using the property: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C%28x%29%29=y ALIGN=MIDDLE  ALT=\"log%28b%2C%28x%29%29=y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ====> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Ey=x ALIGN=MIDDLE  ALT=\"b%5Ey=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10%5E%285%5Ex%29=y ALIGN=MIDDLE  ALT=\"10%5E%285%5Ex%29=y\"   DISABLED_style= \"max-width:90%; height:auto;\" > Rewrite the equation again using the property: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=log%28b%2C%28x%29%29=y ALIGN=MIDDLE  ALT=\"log%28b%2C%28x%29%29=y\"   DISABLED_style= \"max-width:90%; height:auto;\" > ====> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=b%5Ey=x ALIGN=MIDDLE  ALT=\"b%5Ey=x\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So the inverse function is <IMG STYLE=\"max-width:80%;\" SRC=\"/cgi-bin/rendertex.cgi?\LARGE f^{-1}(x)=10^^{5^{x}}\">\r<BR>\n" );
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