document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1009730>Question 1009730</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A WebWork questions has asked if the following functions have inverses:\r<BR>\n" );
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document.write( "1. ln(x^18)<BR>\n" );
document.write( "2. 18ln(x)\r<BR>\n" );
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document.write( "As far as I know, they're both the same because the first can be re-written as the second according to the log exponent property. However, WW claims that the first one does not have an inverse, while the second one does.\r<BR>\n" );
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document.write( "I know that 18 is even, so ln(x^18) is defined for x<0 (where ln(x) isn't defined) and has two parts that are symmetric about the y-axis (so the inverse function wouldn't pass the vertical line test).\r<BR>\n" );
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document.write( "That said, If ln isn't defined for x<0, how can there even be a function for ln(x^18)? Furthermore, isn't the second function (f(x)=18ln(x)) just an alternate form? If I rewrote 18ln(x) to ln(x^18), wouldn't it become undefined as well?\r<BR>\n" );
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document.write( "Or is it that when we start off with ln(x^18), it's implicitly assumed that the value of x can be either negative or positive because a positive exponent means the result will be the same?  In 18ln(x), however, we've started with an exponent of 1, which is odd, so x values will respect their signs.\r<BR>\n" );
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document.write( "But wouldn't this cause some trouble during algebraic manipulation, though? Would we just have to respect the original form? I mean, if I had a situation where I had to isolate x, I would technically be allowed to change ln(x^18) to 18ln(x) and vice versa, right?\r<BR>\n" );
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document.write( "I would really appreciate a clarification on the difference between the two functions. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #625173 by </B><A HREF=/tutors/aboutme.mpl?userid=addingup><B>addingup(3677)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=addingup><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=addingup><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=625173\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> I'm with you, these are two ways of writing the same equation:ln(x^18) or 18ln(x). Again, there may be some dark secret I'm missing, but to me the inverse for either one is the same: e^x/18<BR>\n" );
document.write( "J </SPAN>\n" );
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