Unit 3 · Core unit
Growth by constant ratio, and the inverse that recovers the exponent. Most of the unit follows from those two sentences.
What a strong answer looks like
A strong answer identifies the family from constant ratios and interprets the base as a rate in context.
Know
An exponential multiplies by a constant factor over equal input intervals. A percentage change becomes one plus or minus that rate.
Apply
Convert the stated rate to a factor, then match the exponent to the period the factor describes.
Watch out
Using the percentage itself as the base.
Study move
Write growth and decay models from stated percentages and interpret each base.
Know
Logarithms convert products to sums, quotients to differences, and powers to coefficients, in both directions.
Apply
Expand to separate, or condense to a single logarithm before exponentiating.
Watch out
Splitting the logarithm of a sum, for which no rule exists.
Study move
Expand a logarithm of a product with an exponent and condense it back.
Know
Exponential equations are solved with logarithms; logarithmic equations by exponentiating, which can introduce extraneous roots.
Apply
Check every candidate against the domain requirement that each argument be positive.
Watch out
Skipping the domain check and reporting an impossible root.
Study move
Solve an equation combining two logarithms and reject the invalid root.
Modeling drill
A quantity grows from one known value to another over several years. Write the model, state the annual factor, and interpret it.
