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2026-01-01
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2026-02-28
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<p>Finding points that describe the behavior of a function and drawing a curve through those points is known as<a>graphing</a>a logarithmic function. Depending on the base value, the curve can either increase or decrease.</p>
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<p>Finding points that describe the behavior of a function and drawing a curve through those points is known as<a>graphing</a>a logarithmic function. Depending on the base value, the curve can either increase or decrease.</p>
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<p>The curve increases if the base is greater than 1 \( (base>1)\), and decreases if the base lies between 0 and 1 \((0 < \text{base} < 1)\). To graph a logarithmic function, we must follow certain steps:</p>
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<p>The curve increases if the base is greater than 1 \( (base>1)\), and decreases if the base lies between 0 and 1 \((0 < \text{base} < 1)\). To graph a logarithmic function, we must follow certain steps:</p>
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<p><strong>Step 1:</strong>Identify the domain and range.</p>
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<p><strong>Step 1:</strong>Identify the domain and range.</p>
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<p><strong>Step 2:</strong>Find the vertical asymptote by setting the argument equal to 0. Remember that a logarithmic graph has a vertical asymptote but no horizontal asymptote.</p>
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<p><strong>Step 2:</strong>Find the vertical asymptote by setting the argument equal to 0. Remember that a logarithmic graph has a vertical asymptote but no horizontal asymptote.</p>
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<p><strong>Step 3:</strong>Set the argument equal to 1 by substituting the value of x. To find the x-intercept, use the property loga(1) = 0.</p>
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<p><strong>Step 3:</strong>Set the argument equal to 1 by substituting the value of x. To find the x-intercept, use the property loga(1) = 0.</p>
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<p><strong>Step 4:</strong>Set the argument equal to the base by substituting the value of x. To find another point on the graph, use the property loga(a) = 1.</p>
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<p><strong>Step 4:</strong>Set the argument equal to the base by substituting the value of x. To find another point on the graph, use the property loga(a) = 1.</p>
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<p><strong>Step 5:</strong>Draw the curve by connecting the two points and extending the curve toward the vertical asymptote.</p>
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<p><strong>Step 5:</strong>Draw the curve by connecting the two points and extending the curve toward the vertical asymptote.</p>
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<p>Let’s look at an example to make it easier to understand.</p>
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<p>Let’s look at an example to make it easier to understand.</p>
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<p>Consider the logarithmic function:</p>
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<p>Consider the logarithmic function:</p>
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<p>\(f(x) = \log_{2}(x - 1)\)</p>
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<p>\(f(x) = \log_{2}(x - 1)\)</p>
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<p><strong>Step 1:</strong>The basic form of the logarithmic function is \(f(x) = \log_a(x).\)</p>
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<p><strong>Step 1:</strong>The basic form of the logarithmic function is \(f(x) = \log_a(x).\)</p>
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<p>Here, the base is a=2. Since 2>1, the curve will increase.</p>
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<p>Here, the base is a=2. Since 2>1, the curve will increase.</p>
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<p><strong>Step 2:</strong>Now set the argument greater than 0:</p>
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<p><strong>Step 2:</strong>Now set the argument greater than 0:</p>
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<p>\(x - 1 > 0 \quad \Rightarrow \quad x > 1\)</p>
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<p>\(x - 1 > 0 \quad \Rightarrow \quad x > 1\)</p>
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<p>So, the domain is (1,∞).</p>
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<p>So, the domain is (1,∞).</p>
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<p>\(Range = R.\)</p>
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<p>\(Range = R.\)</p>
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<p><strong>Step 3:</strong>Find the vertical asymptote by setting the argument equal to 0. In a logarithmic function, the argument must be positive (> 0).</p>
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<p><strong>Step 3:</strong>Find the vertical asymptote by setting the argument equal to 0. In a logarithmic function, the argument must be positive (> 0).</p>
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<p>\(x-1=0⇒x=1\)</p>
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<p>\(x-1=0⇒x=1\)</p>
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<p>So, the vertical asymptote is at x=1.</p>
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<p>So, the vertical asymptote is at x=1.</p>
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<p><strong>Step 4:</strong>Next, let’s find some points.</p>
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<p><strong>Step 4:</strong>Next, let’s find some points.</p>
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<p>At x = 2:</p>
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<p>At x = 2:</p>
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<p>\(f(2) = \log_{2}(2 - 1) = \log_{2}(1) = 0\)</p>
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<p>\(f(2) = \log_{2}(2 - 1) = \log_{2}(1) = 0\)</p>
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<p>So, the point is (2,0).</p>
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<p>So, the point is (2,0).</p>
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<p>At x = 3:</p>
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<p>At x = 3:</p>
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<p>\(f(3) = \log_{2}(3 - 1) = \log_{2}(2) = 1\)</p>
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<p>\(f(3) = \log_{2}(3 - 1) = \log_{2}(2) = 1\)</p>
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<p>So, the point is (3,1).</p>
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<p>So, the point is (3,1).</p>
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<p><strong>Step 5:</strong>Draw the graph by connecting the points, starting near the asymptote x=1.</p>
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<p><strong>Step 5:</strong>Draw the graph by connecting the points, starting near the asymptote x=1.</p>
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<p>Here, the curve passes through the points (2,0) and (3, 1), and increases gradually. The red line shows the vertical asymptote at x=1.</p>
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<p>Here, the curve passes through the points (2,0) and (3, 1), and increases gradually. The red line shows the vertical asymptote at x=1.</p>
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