Logarithmic Functions
The log graph is the exponential graph reflected across y = x, and every feature carries over in mirror image. Domain (0, infinity), range all reals, anchor points (1, 0) and (b, 1), and a vertical asymptote at x = 0 where the outputs dive without bound. For b greater than 1 the curve is increasing and concave down: it flattens forever without ever leveling off, because there is no horizontal asymptote and no ceiling.
The mistakes are graph myths: giving the log a ceiling because it flattens, parking the asymptote at x = 1 where the graph merely crosses the axis, evaluating log(0) as 0, and swapping the roles of base, input, and output when converting between forms. Each one is named and drilled in the lesson.
The work
3 ways in · any order
Lesson
Logarithmic Functions
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The log curve climbs forever, just slowly, and its only wall is the vertical one at x = 0. The lesson anchors the graph on (1, 0) and (b, 1), kills the ceiling myth, and closes with ten scenarios reading domains, asymptotes, and long-run behavior.
Diagnostic
10-item topic check
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Ten items spanning the two Topic 2.11 misconceptions: giving the log graph a horizontal ceiling or misplaced wall, and scrambling the roles of base, argument, and output. The bank ships with the Unit 2 data drop.
Targeted Practice
Drill a single misconception
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Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.