Inverses of Exponential Functions
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalThe logarithm base b is the exponential base b run backward: every point (a, c) on the exponential curve reappears as (c, a) on the log curve, the graphs are reflections over the line y = x, and the cancellation identities say the round trip returns the input. The exponential domain (all reals) and range (positive reals) swap wholesale, which is exactly why a log accepts only positive inputs, and the horizontal asymptote y = 0 turns into the log's vertical asymptote x = 0.
The misconceptions are inherited from 2.8 and 2.9 and sharpened: the log misread as one over the exponential, swapped features half-applied (a log given the exponential's domain, or a horizontal asymptote it does not have), and base-exponent roles drifting inside the cancellation identities. All three get drilled with recognition-speed evaluations.
The work
3 ways in · any order
Lesson
Inverses of Exponential Functions
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One pairing, two directions: b to the x sends exponents to powers, and log base b sends them back. The lesson builds the reflected graph with its swapped domain, range, anchors, and asymptote, and drills the cancellation identities to recognition speed. Ten scenarios: identify inverses, evaluate round trips, and place every swapped feature.
Diagnostic
10-item topic check
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Ten items spanning the three Topic 2.10 misconceptions: the log misread as a reciprocal, swapped domains and asymptotes half-applied, and role confusion inside the cancellation identities. Results route you to the drills that fix what fired.
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 it for now and moves you to the next. Two in a row is a checkpoint, not proof: if the error resurfaces later, the misconception comes back.