chap5-summary.xml

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<section xml:id="chap5-summary" xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Chapter Summary and Review</title>
<subsection><title>Key Concepts</title>
<p>
	<ol label="1">
		<li><p><xref ref="assemblage-inverse-functions">Inverse Functions</xref>
		</p></li>
		<li><p>
			We can make a table of values for the inverse function, <m>f^{-1}</m>, by interchanging the columns of a table for <m>f</m>.
		</p></li>
		<li><p>
			If a function is defined by a formula in the form <m>y = f (x)</m>, we can find a formula for its inverse function by solving the equation for <m>x</m> to get <m>x = f^{-1}(y)</m>.
		</p></li>
		<li><p>
			The inverse function <m>f^{-1}</m> undoes the effect of the function <m>f</m>, that is, if we apply the inverse function to the output of <m>f</m>, we return to the original input value.
		</p></li>
		<li><p>
			If <m>f^{-1}</m> is the inverse function for <m>f</m>, then <m>f</m> is also the inverse function for <m>f^{ -1}</m>.
		</p></li>
		<li><p>
			The graphs of <m>f</m> and its inverse function are <term>symmetric about the line <m>y = x</m></term>.
		</p></li>
		<li><p>
			<term>Horizontal line test</term>: If no horizontal line intersects the graph of a function more than once, then the inverse is also a function.
		</p></li>
		<li><p>
			A function that passes the horizontal line test is called <term>one-to-one</term>.
		</p></li>
		<li><p>The inverse of a function <m>f</m> is also a function if and only if <m>f</m> is one-to-one.</p></li>
		<li><p>We define the logarithmic function, <m>g(x) = \log_b x</m>, which takes the log base <m>b</m> of its input values. The log function <m>g(x) = \log_b x</m> is the inverse of the exponential function <m>f (x) = b^x</m>.</p></li>
		<li><p>
			<assemblage><p>
			Because <m>f (x) = b^x</m> and <m>g(x) = \log_b x</m> are inverse functions for <m>b\gt 0, ~b\ne 1</m>,
			<me>\log_b b^x = x\text{, for all }x\text{  and  }~~~b^{\log_b x} = x\text{, for }x\gt 0</me>

			</p></assemblage>
		</p></li>
		<li><p><xref ref="assemblage-log-functions">Logarithmic Functions <m>y = \log_b x</m></xref></p></li>
		<li><p>A <term>logarithmic equation</term> is one where the variable appears inside of a logarithm. We can solve logarithmic equations by converting to exponential form.</p></li>
		<li><p><xref ref="assemblage-solving-log-equations">Steps for Solving Logarithmic Equations</xref></p></li>
		<li><p>The natural base is an irrational number called <m>e</m>, where
		<me>e\approx 2.71828182845</me>
		</p></li>
		<li><p>The <term>natural exponential function</term> is the function <m>f (x) = e^x</m>. The <term>natural log function</term> is the function <m>g(x) = \ln x = \log_e x</m>.</p></li>
		<li><p>
			<xref ref="assemblage-conversion-natural-logs">Conversion Formulas for Natural Logs</xref>
		</p></li>
		<li><p>
			<xref ref="assemblage-properties-natural-logs">Properties of Natural Logarithms</xref>
		</p></li>
		<li><p>We use the natural logarithm to solve exponential equations with base <m>e</m>.</p></li>
		<li><p>
			<xref ref="assemblage-exponential-growth-and-decay">Exponential Growth and Decay</xref>
		</p></li>
		<li><p>
			<term>Continuous compounding</term>: The amount accumulated in an account after <m>t</m> years at interest rate <m>r</m> compounded continuously is given by
			<me>A(t) = Pe^{rt}</me>
			where <m>P</m> is the principal invested.
		</p></li>
		<li><p>A <term>log scale</term> is useful for plotting values that vary greatly in magnitude. We plot the log of the variable, instead of the variable itself.</p></li>
		<li><p>A log scale is a <term>multiplicative scale</term>: Each increment of equal length on the scale indicates that the value is multiplied by an equal amount.</p></li>
		<li><p>The pH value of a substance is defined by the formula
			<me>\text{pH}=-\log_{10}[H^+]</me>
			where <m>[H^+]</m> denotes the concentration of hydrogen ions in the substance.</p></li>
		<li><p>
			The loudness of a sound is measured in decibels, <m>D</m>, by
			<me>D=10 \log_{10}\left(\frac{I}{10^{-12}}\right)</me>
			where <m>I</m> is the intensity of its sound waves (in watts per square meter).
		</p></li>
		<li><p>
			The Richter magnitude, <m>M</m>, of an earthquake is given by
			<me>M=\log_{10}\left(\frac{A}{A_0} \right)</me>
			where <m>A</m> is the amplitude of its seismographic trace and <m>A_0</m> is the amplitude of the smallest detectable earthquake.
		</p></li>
		<li><p>A <em>difference</em> of <m>K</m> units on a logarithmic scale corresponds to a <em>factor</em> of <m>10^K</m> units in the value of the variable.</p></li>
	</ol>.
</p>
</subsection>
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</section>
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