Problem

Using base \(e\) and unit vertical scale, write the logarithmic equation with vertical asymptote \(x~=~2\) that passes through \((3,~0)\). State its domain and verify the asymptote, point, and scale.

Prompt logarithmic graph showing the stated base, unit-scale branch, vertical asymptote, and point; the equation is withheld.

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What this problem is really about

Write the natural-log model with unit vertical scale, a horizontal shift inside the argument, and a possible vertical shift outside. The vertical asymptote fixes the horizontal-shift parameter; substituting the given point then determines the remaining shift. Require the completed argument to be positive, and verify that the point, domain boundary, and unit coefficient all agree.

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Equation: y = ln(x − 2). Asymptote: x = 2. Domain: x > 2. Point check: y(3) = ln(1) = 0. Vertical scale: 1.

Equation: y = ln(x + 2). Asymptote: x = −2. Domain: x > −2. Point check: y(3) = ln(5), not 0. Vertical scale: 1.

Equation: y = 2ln(x − 2). Asymptote: x = 2. Domain: x > 2. Point check: y(3) = 0. Vertical scale: 2, not the stated unit scale.

Equation: y = ln(x − 2) + 1. Asymptote: x = 2. Domain: x > 2. Point check: y(3) = 1, not 0. Vertical scale: 1.

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Using base \(e\) and unit vertical scale, write the logarithmic equation with vertical asymptote \(x~=~2\) that passes through \((3,~0)\). State its domain and verify the asymptote, point, and scale.

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