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<!DOCTYPE html>
<html lang="en">
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<title>Exponential and Logarithmic Functions</title>
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<body>
<header>
Exponential and Logarithmic Functions
</header>
<section>
<h1>Exponential Functions</h1>
<p><strong>Definition:</strong></p>
<p>An <strong>exponential function</strong> is a mathematical function of the form:</p>
<div class="math-box">
<p><span class="formula">f(x) = a · b<sup>x</sup></span></p>
</div>
<p>Where:</p>
<ul>
<li><strong>a</strong> is a constant (the initial value or scaling factor),</li>
<li><strong>b</strong> is the base of the exponential function (where b > 0 and b ≠ 1),</li>
<li><strong>x</strong> is the exponent or power.</li>
</ul>
<p>The exponential function describes exponential growth or decay, depending on the base <strong>b</strong>. If <strong>b > 1</strong>, the function represents exponential growth; if <strong>0 < b < 1</strong>, it represents exponential decay.</p>
<h2>Key Properties of Exponential Functions:</h2>
<ul>
<li><strong>Domain:</strong> The domain of the exponential function is all real numbers ( -∞ < x < ∞ ).</li>
<li><strong>Range:</strong> The range is ( 0, ∞ ) for exponential growth (b > 1) and ( -∞, 0 ) for exponential decay ( 0 < b < 1 ).</li>
<li><strong>Y-intercept:</strong> The graph of the exponential function always passes through the point (0, a) since f(0) = a · b<sup>0</sup> = a.</li>
<li><strong>Horizontal Asymptote:</strong> The line y = 0 is a horizontal asymptote of the graph.</li>
</ul>
<p>Example of Exponential Growth: The population of a species growing at a constant rate can be modeled using the exponential function:</p>
<div class="example">
<p><span class="formula">P(t) = P<sub>0</sub> · e<sup>rt</sup></span></p>
<p>Where:</p>
<ul>
<li><strong>P(t)</strong> is the population at time <strong>t</strong>,</li>
<li><strong>P<sub>0</sub></strong> is the initial population,</li>
<li><strong>r</strong> is the growth rate,</li>
<li><strong>e</strong> is the base of the natural logarithm (approximately 2.71828).</li>
</ul>
</div>
<h1>Logarithmic Functions</h1>
<p><strong>Definition:</strong></p>
<p>A <strong>logarithmic function</strong> is the inverse of an exponential function. It is defined as:</p>
<div class="math-box">
<p><span class="formula">f(x) = log<sub>b</sub>(x)</span></p>
</div>
<p>Where:</p>
<ul>
<li><strong>b</strong> is the base of the logarithm (where <strong>b > 0</strong> and <strong>b ≠ 1</strong>),</li>
<li><strong>x</strong> is the argument of the logarithm, and</li>
<li><strong>f(x)</strong> is the logarithm of <strong>x</strong> to the base <strong>b</strong>.</li>
</ul>
<h2>Properties of Logarithmic Functions:</h2>
<ul>
<li><strong>Domain:</strong> The domain is ( 0, ∞ ) for all logarithmic functions.</li>
<li><strong>Range:</strong> The range is ( -∞, ∞ ) for all logarithmic functions.</li>
<li><strong>Intercept:</strong> The logarithmic function f(x) = log<sub>b</sub>(x) passes through (1, 0) because log<sub>b</sub>(1) = 0.</li>
<li><strong>Asymptote:</strong> The vertical line x = 0 is a vertical asymptote of the graph.</li>
</ul>
<p>Logarithmic Identity and Properties:</p>
<ul>
<li><strong>Change of Base Formula:</strong> <span class="formula">log<sub>b</sub>(x) = log<sub>k</sub>(x) / log<sub>k</sub>(b)</span></li>
<li><strong>Product Rule:</strong> <span class="formula">log<sub>b</sub>(xy) = log<sub>b</sub>(x) + log<sub>b</sub>(y)</span></li>
<li><strong>Quotient Rule:</strong> <span class="formula">log<sub>b</sub>(x/y) = log<sub>b</sub>(x) - log<sub>b</sub>(y)</span></li>
<li><strong>Power Rule:</strong> <span class="formula">log<sub>b</sub>(x<sup>r</sup>) = r log<sub>b</sub>(x)</span></li>
<li><strong>Logarithm of 1:</strong> <span class="formula">log<sub>b</sub>(1) = 0</span></li>
<li><strong>Logarithm of the Base:</strong> <span class="formula">log<sub>b</sub>(b) = 1</span></li>
</ul>
<h1>Applications of Exponential and Logarithmic Functions</h1>
<p><strong>Exponential Growth and Decay:</strong> Population growth, radioactive decay, compound interest, and many physical processes follow exponential models.</p>
<div class="important">
<p>Formula for exponential decay (e.g., for half-life problems):</p>
<p><span class="formula">N(t) = N<sub>0</sub> · e<sup>-kt</sup></span></p>
<p>Where N(t) is the quantity at time t, N<sub>0</sub> is the initial quantity, and k is the decay constant.</p>
</div>
<p><strong>Logarithmic Applications:</strong> Logarithms are widely used in solving problems involving exponential growth and decay.</p>
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