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<b>Data Structures and Algorithms
with Object-Oriented Design Patterns in C++</b><br>
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<H2><A NAME="SECTION004150000000000000000">About Logarithms</A></H2>
<P>
In this section we determine the asymptotic behavior of logarithms.
Interestingly,
despite the fact that <IMG WIDTH=33 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59565" SRC="img353.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img353.gif" > diverges as <I>n</I> gets large,
<IMG WIDTH=63 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59569" SRC="img354.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img354.gif" > for all integers <IMG WIDTH=38 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59063" SRC="img241.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img241.gif" >. Hence, <IMG WIDTH=88 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59573" SRC="img355.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img355.gif" >.
Furthermore, as the following theorem will show,
<IMG WIDTH=33 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59565" SRC="img353.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img353.gif" > raised to any integer power <IMG WIDTH=36 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline59577" SRC="img356.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img356.gif" > is still <I>O</I>(<I>n</I>).
<P>
<BLOCKQUOTE> <b>Theorem</b><A NAME="theoremvii"> </A>
For every integer <IMG WIDTH=36 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline59577" SRC="img356.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img356.gif" >, <IMG WIDTH=96 HEIGHT=30 ALIGN=MIDDLE ALT="tex2html_wrap_inline59583" SRC="img357.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img357.gif" >.
</BLOCKQUOTE>
<P>
extbfProof
This result follows immediately from Theorem <A HREF="page60.html#theoremv" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page60.html#theoremv"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>
and the observation that for all integers <IMG WIDTH=36 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline59577" SRC="img356.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img356.gif" >,
<P><A NAME="eqnloglimit"> </A> <IMG WIDTH=500 HEIGHT=36 ALIGN=BOTTOM ALT="equation1536" SRC="img358.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img358.gif" ><P>
This observation can be proved by induction as follows:
<P>
<b>Base Case</b>
Consider the limit
<P> <IMG WIDTH=289 HEIGHT=36 ALIGN=BOTTOM ALT="displaymath59561" SRC="img359.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img359.gif" ><P>
for the case <I>k</I>=1.
Using L'Hôpital's rule<A NAME="tex2html56" HREF="footnode.html#1686" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/footnode.html#1686"><IMG ALIGN=BOTTOM ALT="gif" SRC="foot_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/foot_motif.gif"></A><A NAME=1555> </A>
we see that
<P> <IMG WIDTH=500 HEIGHT=50 ALIGN=BOTTOM ALT="eqnarray1556" SRC="img365.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img365.gif" ><P>
<P>
<b>Inductive Hypothesis</b>
Assume that Equation <A HREF="page62.html#eqnloglimit" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page62.html#eqnloglimit"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A> holds for <IMG WIDTH=104 HEIGHT=22 ALIGN=MIDDLE ALT="tex2html_wrap_inline59607" SRC="img366.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img366.gif" >.
Consider the case <I>k</I>=<I>m</I>+1.
Using L'Hôpital's rule<A NAME=1567> </A> we see that
<P> <IMG WIDTH=500 HEIGHT=94 ALIGN=BOTTOM ALT="eqnarray1568" SRC="img367.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img367.gif" ><P>
<P>
Therefore, by induction on <I>m</I>, Equation <A HREF="page62.html#eqnloglimit" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page62.html#eqnloglimit"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>
holds for all integers <IMG WIDTH=36 HEIGHT=26 ALIGN=MIDDLE ALT="tex2html_wrap_inline59577" SRC="img356.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img356.gif" >.
<P>
For example,
using this property of logarithms
together with the rule for determining the asymptotic behavior
of the product of two functions (Theorem <A HREF="page60.html#theoremiii" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/page60.html#theoremiii"><IMG ALIGN=BOTTOM ALT="gif" SRC="cross_ref_motif.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/icons/cross_ref_motif.gif"></A>),
we can determine that since <IMG WIDTH=88 HEIGHT=24 ALIGN=MIDDLE ALT="tex2html_wrap_inline59573" SRC="img355.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img355.gif" >,
then <IMG WIDTH=107 HEIGHT=25 ALIGN=MIDDLE ALT="tex2html_wrap_inline59617" SRC="img368.gif" tppabs="http://dictator.uwaterloo.ca/Bruno.Preiss/books/opus4/html/img368.gif" >.
<P>
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