Analyze graph transformations across radical, rational, exponential, logarithmic, and other functions; recognize even and odd functions.
Analyze a transformed logarithm and its domain boundary
Problem
Analyze \(y~=~\ln(x~-~2)~+~3\) relative to \(y~=~\ln(x)\). State the base and transformations, vertical asymptote, domain, and image of parent point \((1,~0)\).
Compare the shifted asymptote and moved key point.Open full size
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What this problem is really about
A logarithm is organized around an excluded input boundary. Set its argument equal to zero to locate that boundary, then use the strict positivity requirement to determine which side belongs to the domain. Map the parent point where the logarithm’s output is zero as an independent check, remembering that an outside shift changes outputs but not the boundary input.
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