Logarithmic Function Manipulation
▶︎ Watch it animatedinteractive step-through · ~3 min · optionalEvery log property is an exponent rule in disguise, and there are exactly three: the log of a product adds, the log of a quotient subtracts, and an exponent inside a log slides out front as a coefficient. Expanding runs the rules forward to isolate variables; condensing runs them backward to set up equation solving. Change of base, log_b(x) = log(x) / log(b), is a quotient of two separate logs, a different animal from the log of a quotient.
The mistakes live in the fake properties: splitting log(x + y) into log x + log y, multiplying logs where the product rule says add, canceling logs across a fraction, and letting a coefficient merge before it becomes an exponent. The lesson names each fake, shows the two-line numeric refutation, and drills the real moves.
The work
3 ways in · any order
Lesson
Logarithmic Function Manipulation
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Products add, quotients subtract, powers slide out front: three real properties, all exponent rules in disguise. The lesson drills expanding and condensing, separates change of base from the quotient rule, and closes with ten scenarios spotting the fake properties that do not exist.
Diagnostic
10-item topic check
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Ten items on the Topic 2.12 misconception cluster: log-of-sum splitting, products and quotients of logs invented where the properties say add and subtract, and coefficients merged before becoming exponents. 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.