section-4-2.xml

<?xml version="1.0" encoding="UTF-8" ?>
<!-- <mathbook><book> -->

<section xml:id="Exponential-Functions"   xmlns:xi="http://www.w3.org/2001/XInclude">
<title>Exponential Functions</title>
<introduction>
<p>
	In <xref ref="Exponential-Growth-and-Decay" text="type-global"/>, we studied functions that describe exponential growth or decay. More formally, we define an <term>exponential function</term> as follows,
</p>
<assemblage><title>Exponential Function</title>
<p>
	<me>f(x) = ab^x \text{, where } b \gt 0 \text{ and } b \ne 1 \text{, } a \ne 0</me>
</p></assemblage>
<p>
	Some examples of exponential functions are
	<me>f (x) = 5^x \text{, } P(t) = 250(1.7)^t \text{, and } g(n) = 2.4(0.3)^n</me>
	The constant <m>a</m> is the <m>y</m>-intercept of the graph because
	<me>f (0) = a \cdot b^0 = a \cdot 1 = a</me>
	For the examples above, we find that the <m>y</m>-intercepts are
	\begin{align*}
	f(0) \amp= 5^0 = 1 \text{,} \\
	P(0) \amp= 250(1.7)^0 = 250\text{, and} \\
	g(0) \amp= 2.4(0.3)^0 = 2.4
	\end{align*}
</p>
<p>
	The positive constant <m>b</m> is called the <term>base</term> of the exponential function.</p>

<p><ul>
    <li> We do not allow <m>b</m> to be negative, because if <m>b \lt 0</m>, then <m>b^x</m> is not a real number for some values of <m>x</m>. For example, if <m>b = -4</m> and <m>f (x) = (-4)^x</m>, then <m>f (1/2) = (-4)^{1/2}</m> is an imaginary number.</li>
    <li> We also exclude <m>b = 1</m> as a base because <m>1^x = 1</m> for all values of <m>x</m>; hence the function <m>f (x) = 1^x</m> is actually the constant function <m>f (x) = 1</m>.</li>
</ul>
</p>
</introduction>
<subsection><title>Graphs of Exponential Functions</title>
<p>
    The graphs of exponential functions have two characteristic shapes, depending on whether the base, <m>b</m>, is greater than <m>1</m> or less than <m>1</m>. As typical examples, consider the graphs of <m>f (x) = 2^x</m> and <m>g(x) =\left(\frac{1}{2}\right)^x</m> shown in <xref ref="fig-exponential-growth-and-decay" text="type-global"/>. Some values for <m>f</m> and <m>g</m> are recorded in <xref ref="table-exp-growth">Tables</xref> and <xref ref="table-exp-decay" text="global"/>
</p>
<sidebyside widths="16% 16% 68%">
	<table width="17%" xml:id="table-exp-growth"><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell><m>x</m></cell>
            <cell><m>f(x)</m></cell>
        </row>
        <row>
            <cell><m>-3</m></cell>
            <cell><m>\frac{1}{8}</m></cell>
        </row>
        <row>
            <cell><m>-2</m></cell>
            <cell><m>\frac{1}{4}</m></cell>
        </row>
        <row>
            <cell><m>-1</m></cell>
            <cell><m>\frac{1}{2}</m></cell>
        </row>
        <row>
            <cell><m>0</m></cell>
            <cell><m>1</m></cell>
        </row>
        <row>
            <cell><m>1</m></cell>
            <cell><m>2</m></cell>
        </row>
        <row>
            <cell><m>2</m></cell>
            <cell><m>4</m></cell>
        </row>
        <row>
            <cell><m>3</m></cell>
            <cell><m>8</m></cell>
        </row>
    </tabular><caption></caption></table>

	<table width="17%" xml:id="table-exp-decay"><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell><m>x</m></cell>
            <cell><m>g(x)</m></cell>
        </row>
        <row>
            <cell><m>-3</m></cell>
            <cell><m>8</m></cell>
        </row>
        <row>
            <cell><m>-2</m></cell>
            <cell><m>4</m></cell>
        </row>
        <row>
            <cell><m>-1</m></cell>
            <cell><m>2</m></cell>
        </row>
        <row>
            <cell><m>0</m></cell>
            <cell><m>1</m></cell>
        </row>
        <row>
            <cell><m>1</m></cell>
            <cell><m>\frac{1}{2}</m></cell>
        </row>
        <row>
            <cell><m>2</m></cell>
            <cell><m>\frac{1}{4}</m></cell>
        </row>
        <row>
            <cell><m>3</m></cell>
            <cell><m>\frac{1}{8}</m></cell>
        </row>
    </tabular><caption></caption></table>

    <figure xml:id="fig-exponential-growth-and-decay"><caption></caption><image source="images/fig-exponential-growth-and-decay"><description>graphs of exponential growth and decay</description></image></figure>

</sidebyside>

<p>
	Notice that <m>f (x) = 2^x</m> is an increasing function and <m>g(x) = \left(\dfrac{1}{2}\right)^x</m> is a decreasing function. Both are concave up. In general, exponential functions have the following properties.
</p>

<assemblage><title>Properties of Exponential Functions, <m>f(x) = ab^x</m>, <m>a \gt 0</m></title><p>
<ol>
	<li>
		<p>Domain: all real numbers.</p>
	</li>
	<li>
		<p>Range: all positive numbers.</p>
	</li>
	<li>
		<p>If <m>b \gt 1</m>, the function is increasing and concave up;</p> 
		<p>if <m>0 \lt b \lt 1</m>, the function is decreasing and concave up.</p>
	</li>
	<li>
		<p>The <m>y</m>-intercept is <m>(0, a)</m>. There is no <m>x</m>-intercept.</p>
	</li>
</ol></p>
</assemblage>

<p>
	In <xref ref="table-exp-growth" text="type-global"/> you can see that as the <m>x</m>-values decrease toward negative infinity, the corresponding <m>y</m>-values decrease toward zero. As a result, the graph of <m>f</m> decreases toward the <m>x</m>-axis as we move to the left. Thus, the negative <m>x</m>-axis is a <term>horizontal asymptote</term> for exponential functions with <m>b \gt 1</m>, as shown in <xref ref="fig-exponential-growth-and-decay" text="type-global"/>a.</p>
    <p> For exponential functions with <m>0 \lt b \lt 1</m>, the positive <m>x</m>-axis is an asymptote, as illustrated in <xref ref="fig-exponential-growth-and-decay" text="type-global"/>b. (See <xref ref="basic-functions" text="type-global"/> to review asymptotes.)
</p>
<p>
	In <xref ref="example-two-exponential-growth" text="type-global"/>, we compare two increasing exponential functions. The larger the value of the base, <m>b</m>, the faster the function grows. In this example, both functions have <m>a = 1</m>.
</p>
<example xml:id="example-two-exponential-growth">
<p>
	Compare the graphs of <m>f (x) = 3^x</m> and <m>g(x) = 4^x</m>.
</p>
<solution><p>
	We evaluate each function for several convenient values, as shown in <xref ref="table-two-exponential-growth" text="type-global"/>.
</p>
<p>
	Plot the points for each function and connect them with smooth curves. For positive <m>x</m>-values, <m>g(x)</m> is always larger than <m>f (x)</m>, and is increasing more rapidly. In <xref ref="fig-two-exponential-growth" text="type-global"/>, <m>g(x) = 4^x</m> climbs more rapidly than <m>f (x) = 3^x</m>. Both graphs cross the <m>y</m>-axis at (0, 1).
</p>
<p>
<sidebyside>
<table valign="middle" width="40%" xml:id="table-two-exponential-growth"><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell><m>x</m></cell>
            <cell><m>f(x)</m></cell>
            <cell><m>g(x)</m></cell>
        </row>
        <row>
            <cell><m>-2</m></cell>
            <cell><m>\dfrac{1}{9}</m></cell>
            <cell><m>\dfrac{1}{16}</m></cell>
        </row>
        <row>
            <cell><m>-1</m></cell>
            <cell><m>\dfrac{1}{3}</m></cell>
            <cell><m>\dfrac{1}{4}</m></cell>
        </row>
        <row>
            <cell><m>0</m></cell>
            <cell><m>1</m></cell>
            <cell><m>1</m></cell>
        </row>
        <row>
            <cell><m>1</m></cell>
            <cell><m>3</m></cell>
            <cell><m>4</m></cell>
        </row>
        <row>
            <cell><m>2</m></cell>
            <cell><m>9</m></cell>
            <cell><m>16</m></cell>
        </row>
    </tabular><caption></caption></table>
	<figure valign="middle"  xml:id="fig-two-exponential-growth"><caption></caption><image source="images/fig-two-exponential-growth"  width="60%"><description>two exponential growth functions</description></image></figure>
</sidebyside></p>
</solution>
</example>

<sidebyside widths="50% 35%">
<p valign="top">
	For decreasing exponential functions, those with bases between <m>0</m> and <m>1</m>, the smaller the base, the more steeply the graph decreases. For example, compare the graphs of <m>p(x) = 0.8^x</m> and <m>q(x) = 0.5^x</m> shown in <xref ref="fig-two-exponential-decay" text="type-global"/>.
</p>
<figure xml:id="fig-two-exponential-decay"><caption></caption><image source="images/fig-two-exponential-decay"><description>two exponential decay functions</description></image></figure>
</sidebyside>

<exercise xml:id="exercise-compare-exponential"><statement>
<ol label="a">
	<li>
		State the ranges of the functions <m>f</m> and <m>g</m> in <xref ref="fig-two-exponential-growth" text="type-global"/> from <xref ref="example-two-exponential-growth" text="type-global"/> on the domain <m>[-2, 2]</m>.
	</li>
	<li>
		State the ranges of the functions <m>p</m> and <m>q</m> shown in <xref ref="fig-two-exponential-decay" text="type-global"/> on the domain <m>[-2, 2]</m>. Round your answers to two decimal places.
	</li>
</ol>
</statement></exercise>
</subsection>



<subsection><title>Transformations of Exponential Functions</title>
<p>
	In <xref ref="chap2" text="type-global"/>, we considered transformations of the basic graphs. For instance, the graphs of the functions <m>y = x^2 - 4</m> and <m>y = (x - 4)^2</m> are shifts of the basic parabola, <m>y = x^2</m>. In a similar way, we can shift or stretch the graph of an exponential function while the basic shape is preserved.
</p>
<example xml:id="example-GC-shift-exponentials">
<p>
	Use your calculator to graph the following functions. Describe how these graphs compare with the graph of <m>h(x) = 2^x</m>.
	<ol label="a">
		<li><m>f (x) = 2^x + 3</m></li>
		<li><m>g(x) = 2^{x+3}</m></li>
	</ol>
</p>
<solution>
<p>
	Enter the formulas for the three functions as shown below. Note the parentheses around the exponent in the keying sequence for <m>Y_3 = g(x)</m>
</p>
<p>
	<m>Y_1 = 2 </m> <c>^</c> X 
</p>
<p>
	<m>Y_2 = 2 </m> <c>^</c> X <c>+</c> 3
</p>
	<p>
	<m>Y_3 = 2 </m> <c>^</c> <c>(</c> X <c>+</c> 3 <c>)</c>
</p>
<p>
	The graphs of <m>h(x) = 2^x</m>, <m>f(x) = 2^x + 3</m>, and <m>g(x) = 2^{x+3}</m> are shown using the standard window in <xref ref="fig-GC-exponential-growth" text="type-global"/>
	<figure xml:id="fig-GC-exponential-growth"><caption></caption><image source="images/fig-GC-exponential-growth"  width="100%"><description>GC graphs of transformations of exponential growth</description></image></figure>
<ol label="a">
	<li>
		The graph of <m>f(x) = 2^x + 3</m>, shown in <xref ref="fig-GC-exponential-growth" text="type-global"/>b, has the same basic shape as that of <m>h(x) = 2^x</m>, but it has a horizontal asymptote at <m>y = 3</m> instead of at <m>y = 0</m> (the <m>x</m>-axis). In fact, <m>f(x) = h(x) + 3</m>, so the graph of <m>f</m> is a vertical translation of the graph of <m>h</m> by <m>3</m> units. If every point on the graph of <m>h(x) = 2^x</m> is moved <m>3</m> units upward, the result is the graph of <m>f (x) = 2^x + 3</m>.
	</li>
	<li>
		First note that <m>g(x) = 2^x+3 = h(x + 3)</m>. In fact, the graph of <m>g(x) = 2^{x+3}</m> shown in Figure <xref ref="fig-GC-exponential-growth" text="type-global"/>c has the same basic shape as <m>h(x) = 2^x</m> but has been translated <m>3</m> units to the left.
	</li>
</ol>
</p></solution>
</example>
<p>
	Recall that the graph of <m>y = -f (x)</m> is the reflection about the <m>x</m>-axis of the graph of <m>y = f (x)</m>. The graphs of <m>y = 2^x</m> and <m>y = -2^x</m> are shown in  <xref ref="fig-exponential-vertical-flip" text="type-global"/>. You may have also noticed a relationship between the graphs of <m>f (x) = 2^x</m> and <m>g(x) = \left(\dfrac{1}{2}\right)^x</m>, which are shown in <xref ref="fig-exponential-horizontal-flip" text="type-global"/>.
</p>
	
<sidebyside widths="35% 35%">
	<figure xml:id="fig-exponential-vertical-flip"><caption></caption><image source="images/fig-exponential-vertical-flip"><description>vertical flip of an exponential function</description></image></figure>
	<figure xml:id="fig-exponential-horizontal-flip"><caption></caption><image source="images/fig-exponential-horizontal-flip"><description>horizontal flip of an exponential function</description></image></figure>
</sidebyside>
<p>
	The graph of <m>g</m> is the reflection of the graph of <m>f</m> about the <m>y</m>-axis. We can see why this is true by writing the formula for <m>g(x)</m> in another way:
	<me>g(x) =\left(\frac{1}{2}\right)^x= \left(2^{-1}\right)^x = 2^{-x}</me>
	We see that <m>g(x)</m> is the same function as <m>f (-x)</m>. Replacing <m>x</m> by <m>-x</m> in the formula for a function switches every point <m>( p, q)</m> on the graph with the point <m>(-p, q)</m> and thus reflects the graph about the <m>y</m>-axis.
</p>
<assemblage><title>Reflections of Graphs</title><idx>reflections of graphs</idx><p>
<ol>
	<li>
		<p>The graph of <m>y = -f (x)</m> is the reflection of the graph of <m>y = f (x)</m> about the <m>x</m>-axis.</p>
	</li>
	<li><p>
		The graph of <m>y = f (-x)</m> is the reflection of the graph of <m>y = f (x)</m> about the <m>y</m>-axis.</p>
	</li>
</ol></p>
</assemblage>

<exercise xml:id="exercise-exponential-flips"><statement>
	Which of the functions below have the same graph? Explain why.
	<ol label="a">
		<li><m>f (x) =\left(\dfrac{1}{4}\right)^x</m></li>
		<li><m>g(x) = -4^x</m></li>
		<li><m>h(x) = 4^{-x}</m></li>
	</ol>
</statement></exercise>
</subsection>



<subsection><title>Comparing Exponential and Power Functions</title>
<p>
	Exponential functions<idx>exponential functions</idx> are not the same as the power functions <idx>power functions</idx> we studied in <xref ref="chap3" text="type-global"/>. Although both involve expressions with exponents, it is the location of the variable that makes the difference.
</p>
<assemblage><title>Power Functions vs Exponential Functions</title><p>
<sidebyside>
   <tabular>
        <col halign="left" width="40%"/>
        <col halign="left" width="25%"/>
        <col halign="left" width="25%"/>
         <row>
            <cell><p></p></cell>
            <cell><p><term>Power Functions</term></p></cell>
            <cell><p><term>Exponential Functions</term></p></cell>
        </row>
        <row>
            <cell><p><em>General formula</em></p></cell>
            <cell><p><m>h(x)=kx^p</m></p></cell>
            <cell><p><m>f(x)=ab^x</m></p></cell>
        </row>
        <row>
            <cell><p><em>Description</em></p></cell>
            <cell><p>variable base and constant exponent</p></cell>
            <cell><p>constant base and variable exponent</p></cell>
        </row>
        <row>
            <cell><p><em>Example</em></p></cell>
            <cell><p><m>h(x)=2x^3</m></p></cell>
            <cell><p><m>f(x)=2(3^x)</m></p></cell>
        </row>
    </tabular>
</sidebyside></p>
</assemblage>
<p>
	These two families of functions have very different properties, as well.
</p>
<example xml:id="example-power-vs-exponential">
<p>
	Compare the power function <m>h(x) = 2x^3</m> and the exponential function <m>f(x) = 2(3^x)</m>.
</p>
<solution>
   <p>First, compare the values for these two functions in <xref ref="table-power-vs-exponential" text="type-global"/>.</p>  
<sidebyside>
<p width="55%"><parapgraphs>
	<p>The scaling exponent for <m>h(x)</m> is <m>3</m>, so that when <m>x</m> doubles, say, from <m>1</m> to <m>2</m>, the output is multiplied by <m>2^3</m>, or <m>8</m>.</p>
    <p>On the other hand, we can tell that <m>f</m> is exponential because its values increase by a factor of <m>3</m> for each unit increase in <m>x</m>. (To see this, divide any function value by the previous one.)</p></parapgraphs>
</p>
<table valign="top" xml:id="table-power-vs-exponential"><tabular top="major" halign="center" right="minor" left="minor" bottom="minor">
        <row bottom="minor">
            <cell><m>x</m></cell>
            <cell><m>h(x)=2x^3</m></cell>
            <cell><m>f(x)=2(3^x)</m></cell>
        </row>
        <row>
            <cell><m>-3</m></cell>
            <cell><m>-54</m></cell>
            <cell><m>\dfrac{2}{27}</m></cell>
        </row>
        <row>
            <cell><m>-2</m></cell>
            <cell><m>-16</m></cell>
            <cell><m>\dfrac{1}{4}</m></cell>
        </row>
        <row>
            <cell><m>-1</m></cell>
            <cell><m>-2</m></cell>
            <cell><m>\dfrac{2}{3}</m></cell>
        </row>
        <row>
            <cell><m>0</m></cell>
            <cell><m>0</m></cell>
            <cell><m>2</m></cell>
        </row>
        <row>
            <cell><m>1</m></cell>
            <cell><m>2</m></cell>
            <cell><m>6</m></cell>
        </row>
        <row>
            <cell><m>2</m></cell>
            <cell><m>16</m></cell>
            <cell><m>18</m></cell>
        </row>
        <row>
            <cell><m>3</m></cell>
            <cell><m>54</m></cell>
            <cell><m>54</m></cell>
        </row>
    </tabular><caption></caption></table>
</sidebyside>
<p>
	As you would expect, the graphs of the two functions are also quite different. For starters, note that the power function goes through the origin, while the exponential function has <m>y</m>-intercept <m>(0, 2)</m>. (See <xref ref="fig-power-vs-exponential-1" text="type-global"/>)</p>
    <p> From the table, we see that <m>h(3) = f(3) = 54</m>, so the two graphs intersect at <m>x = 3</m>. (They also intersect at approximately <m>x = 2.48</m>.) However, if you compare the values of <m>h(x) = 2x^3</m> and <m>f(x) = 2(3^x)</m> for larger values of <m>x</m>, you will see that eventually the exponential function overtakes the power function, as shown in <xref ref="fig-power-vs-exponential-2" text="type-global"/>.
</p>
<p>
<sidebyside widths="40% 40%">
<figure valign="top" xml:id="fig-power-vs-exponential-1"><caption></caption><image source="images/fig-power-vs-exponential-1"><description>power function vs exponential from -2 to 3</description></image></figure>
<figure valign="top"  xml:id="fig-power-vs-exponential-2"><caption></caption><image source="images/fig-power-vs-exponential-2"><description>power function vs exponential from 0 to 6</description></image></figure>
</sidebyside></p>
</solution>
</example>

<sidebyside widths="40% 50%">
<figure valign="top"  xml:id="fig-power-vs-exponential-3"><caption></caption><image source="images/fig-power-vs-exponential-3"><description>power function vs exponential from 0 to 50</description></image></figure>
<p>
	The relationship in <xref ref="example-power-vs-exponential" text="type-global"/> holds true for all increasing power and exponential functions: For large enough values of <m>x</m>, the exponential function will always be greater than the power function, regardless of the parameters in the functions. <xref ref="fig-power-vs-exponential-3" text="type-global"/> shows the graphs of <m>f(x) = x^6</m> and <m>g(x) = 1.8^x</m>. At first, <m>f (x) \gt g(x)</m>, but at around <m>x = 37</m>, <m>g(x)</m> overtakes <m>f (x)</m>, and <m>g(x) \gt f (x)</m> for all <m>x \gt 37</m>.
</p>
</sidebyside>

<exercise xml:id="exercise-identify-power-and-exponential-functions"><statement>
Which of the following functions are exponential functions, and which are power functions?
<ol label="a">
	<li><m>F(x) = 1.5^x</m></li>
	<li><m>G(x) = 3x^{1.5}</m></li>
	<li><m>H(x) = 3^{1.5x}</m></li>
	<li><m> K(x) = (3x)^{1.5}</m></li>
</ol>
</statement></exercise>

</subsection>



<subsection><title>Exponential Equations</title>
<p>
	An <term>exponential equation</term><idx>exponential equation</idx> is one in which the variable is part of an exponent. For example, the equation
	<me>3^x = 81</me>
	is exponential. Many exponential equations can be solved by writing both sides of the equation as powers with the same base. To solve the equation above, we write
	<me>3^x = 3^4</me>
	which is true if and only if <m>x = 4</m>. In general, if two equivalent powers have the same base, then their exponents must be equal also, as long as the base is not <m>0</m> or <m>\pm 1</m>.
</p>
<p>
	Sometimes the laws of exponents can be used to express both sides of an equation as single powers of a common base.
</p>

<example xml:id="example-exponential-equations">
<p>
	Solve the following equations.
	<ol label="a">
		<li><m>3^{x-2} = 9^3</m></li>
		<li><m> 27 \cdot 3^{-2x} = 9^{x+1}</m></li>
	</ol>
</p>
<solution>
<ol label="a">
	<li>
		Using the fact that <m>9 = 3^2</m>, we write each side of the equation as a power of <m>3</m>:
		\begin{align*}
		3^{x-2} \amp = \left(3^2\right)^3 \\
		3^{x-2} \amp = 3^6 
		\end{align*}
		Now we equate the exponents to obtain
		\begin{align*}
		x - 2 \amp = 6 \\
		x  \amp = 8
		\end{align*}
	</li>
	<li>
		We write each factor as a power of <m>3</m>.
		<me>3^3 \cdot 3^{-2x} = \left(3^2\right)^{x+1}</me>
		We use the laws of exponents to simplify each side:
		<me>3^{3-2x} = 3^{2x+2}</me>
		Now we equate the exponents to obtain
		\begin{align*}
		3 - 2x \amp = 2x + 2 \\
		-4x =\amp  -1
		\end{align*}
		The solution is <m>x = \dfrac{1}{4}</m>.
	</li>
</ol></solution>
</example>

<exercise xml:id="exercise-exponential-equation"><statement>
<p>Solve the equation <m>2^{x+2} = 128</m>.</p>
</statement>
<hint>
    <p>
        Write each side as a power of <m>2</m>.
    </p>
    <p>
        Equate exponents.
    </p>
</hint>
</exercise>

<example xml:id="example-flea-population">
	<p>
		During the summer a population of fleas doubles in number every <m>5</m> days. If a population starts with <m>10</m> fleas, how long will it be before there are <m>10,240</m> fleas?
	</p>
<solution>
<p>
	Let <m>P</m> represent the number of fleas present after <m>t</m> days. The original population of <m>10</m> is multiplied by a factor of <m>2</m> every <m>5</m> days, or
	<me>P(t) = 10 \cdot 2^{t/5}</me>
	We set <m>P = \alert{10,240}</m> and solve for <m>t</m>:
	\begin{align*}
	\alert{10,240} \amp = 10\cdot 2^{t/5}\amp\amp \text{Divide both sides by 10.} \\
	1024 \amp = 2^{t/5} \amp\amp \text {Write 1024 as a power of 2.} \\
	2^{10} \amp = 2^{t/5}
	\end{align*}
	We equate the exponents to get <m>10 = \dfrac{t}{5}</m>, or <m>t = 50</m>. The population will grow to <m>10,240</m> fleas in <m>50</m> days.
</p>
</solution></example>

<exercise xml:id="exercise-ad-campaign"><statement>
<p>
	During an advertising campaign in a large city, the makers of Chip-O’s corn chips estimate that the number of people who have heard of Chip-O’s increases by a factor of <m>8</m> every 4 days.
	<ol label="a">
		<li>
			If 100 people are given trial bags of Chip-O's to start the campaign, write a function, <m>N(t)</m>, for the number of people who have heard of Chip-O's after <m>t</m> days of advertising.
		</li>
		<li>
			Use your calculator to graph the function <m>N(t)</m> on the domain <m>0 \le t \le 15</m>.
		</li>
		<li>
			How many days should the makers run the campaign in order for Chip-O's to be familiar to <m>51,200</m> people? Use algebraic methods to find your answer and verify on your graph.
		</li>
	</ol>
</p>
</statement></exercise>

<technology><title>Graphical Solution of Exponential Equations</title>
<p>
	It is not always so easy to express both sides of the equation as powers of the same base. In the following sections, we will develop more general methods for finding exact solutions to exponential equations. But we can use a graphing calculator to obtain approximate solutions.
</p>

<example xml:id="example-gc-exponential-equation">
<p>
	Use the graph of <m>y = 2^x</m> to find an approximate solution to the equation <m>2^x = 5</m> accurate to the nearest hundredth.
</p>
<solution>
<p>
	Enter <m>Y_1 = 2</m> <c>^</c> X and use the standard graphing window (<c>ZOOM</c> 6) to obtain the graph shown in <xref ref="fig-GC-exponential-equation" text="type-global"/>a. We are looking for a point on this graph with <m>y</m>-coordinate <m>5</m>. Using the TRACE feature, we see that the <m>y</m>-coordinates are too small when <m>x \lt 2.1</m> and too large when <m>x \gt 2.4</m>. 
</p>
<figure xml:id="fig-GC-exponential-equation"><caption></caption><image source="images/fig-GC-exponential-equation"  width="100%"><description>two GC displays of exponential function</description></image></figure>
<p>
	The solution we want lies somewhere between <m>x = 2.1</m> and <m>x = 2.4</m>, but this approximation is not accurate enough. To improve our approximation, we will use the <term>intersect</term><idx>intersect</idx> feature. Set <m>Y_2 = 5</m> and press <c>GRAPH</c>. The <m>x</m>-coordinate of the intersection point of the two graphs is the solution of the equation <m>2^x = 5</m> Activating the <term>intersect</term> command results in <xref ref="fig-GC-exponential-equation" text="type-global"/>b, and we see that, to the nearest hundredth, the solution is <m>2.32</m>.
</p>
<p>
	We can verify that our estimate is reasonable by substituting into the equation:
	<me>2^{2.32}  \stackrel{?}{=} 5</me>
	We enter 2 <c>^</c> 2.32 <c>ENTER</c> to get <m>4.993322196</m>. This number is not equal to <m>5</m>, but it is close, so we believe that <m>x = 2.32</m> is a reasonable approximation to the solution of the equation <m>2^x = 5</m>.
</p>
 </solution>
</example>
</technology>

<exercise xml:id="exercise-GC-exponential-equation"><statement>
	Use the graph of <m>y = 5^x</m> to find an approximate solution to <m>5^x = 285</m>, accurate to two decimal places.
</statement></exercise>


</subsection>
<xi:include href="./summary-4-2.xml" /> <!-- summary  -->
<xi:include href="./section-4-2-exercises.xml" /> <!-- exercises  -->
</section> 
<!-- </book>  </mathbook> -->