103 lines
2.6 KiB
TeX
103 lines
2.6 KiB
TeX
\section{Logarithms}
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\definition{}<logdef>
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The \textit{logarithm} is the inverse of the exponent. That is, if $b^p = c$, then $\log_b{c} = p$. \\
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In other words, $\log_b{c}$ asks the question ``what power do I need to raise $b$ to to get $c$?'' \\
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\medskip
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In both $b^p$ and $\log_b{c}$, the number $b$ is called the \textit{base}.
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\problem{}
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Evaluate the following by hand:
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\begin{enumerate}
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\item $\log_{10}{(1000)}$
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\vfill
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\item $\log_2{(64)}$
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\vfill
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\item $\log_2{(\frac{1}{4})}$
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\vfill
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\item $\log_x{(x)}$ for any $x$
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\vfill
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\item $log_x{(1)}$ for any $x$
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\vfill
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\end{enumerate}
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\pagebreak
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\definition{}
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There are a few ways to write logarithms:
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\begin{itemize}
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\item[] $\log{x} = \log_{10}{x}$
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\item[] $\lg{x} = \log_{10}{x}$
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\item[] $\ln{x} = \log_e{x}$
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\end{itemize}
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\definition{}
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The \textit{domain} of a function is the set of values it can take as inputs. \\
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The \textit{range} of a function is the set of values it can produce.
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\medskip
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For example, the domain and range of $f(x) = x$ is $\mathbb{R}$, all real numbers. \\
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The domain of $f(x) = |x|$ is $\mathbb{R}$, and its range is $\mathbb{R}^+ \cup \{0\}$, all positive real numbers and 0. \\
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\medskip
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Note that the domain and range of a function are not always equal.
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\problem{}<expdomain>
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What is the domain of $f(x) = 5^x$? \\
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What is the range of $f(x) = 5^x$?
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\vfill
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\problem{}<logdomain>
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What is the domain of $f(x) = \log{x}$? \\
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What is the range of $f(x) = \log{x}$?
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\vfill
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\pagebreak
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\problem{}<logids>
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Prove the following identities: \\
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\begin{enumerate}[itemsep=2mm]
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\item $\log_b{(b^x)} = x$
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\item $b^{\log_b{x}} = x$
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\item $\log_b{(xy)} = \log_b{(x)} + \log_b{(y)}$
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\item $\log_b{(\frac{x}{y})} = \log_b{(x)} - \log_b{(y)}$
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\item $\log_b{(x^y)} = y \log_b{(x)}$
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\end{enumerate}
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\vfill
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\begin{instructornote}
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A good intro to the following sections is the linear slide rule:
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\begin{center}
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\begin{tikzpicture}[scale=1]
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\linearscale{2}{1}{}
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\linearscale{0}{0}{}
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\slideruleind
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{5}
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{1}
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{2 + 3 = 5}
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\end{tikzpicture}
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\end{center}
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Take two linear rulers, offset one, and you add. \\
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If you do the same with a log scale, you multiply! \\
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\vspace{1ex}
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Note that the slide rules above start at 0.
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\linehack{}
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After assembling the paper slide rule, you can make a visor with some transparent tape. Wrap a strip around the slide rule, sticky side out, and stick it to itself to form a ring. Cover the sticky side with another layer of tape, and trim the edges to make them straight. Use the edge of the visor to read your slide rule!
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\end{instructornote}
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\pagebreak |