Unit 2
A short unit with a single organising idea: exponential functions change by equal ratios over equal input intervals, and logarithms are the inverse that recovers the exponent. Most of the unit is consequences of that sentence, including modeling, equation solving, and the semi-log plot.
Lesson 2.1
At equal input increments, constant output differences suggest a linear model and constant positive output ratios suggest an exponential model. Check spacing before comparing ratios. A finite data set supports a candidate model rather than proving a unique rule.
Lesson 2.2
A percent increase uses a factor one plus the rate; a percent decrease uses one minus the rate. The rate must match the time unit in the exponent. Repeated percent change multiplies rather than adds.
Lesson 2.3
An exponent counts how many growth periods have elapsed. If a quantity doubles every d time units, use time divided by d. Equivalent models can express the same growth with different time units or bases.
Lesson 2.4
A logarithm answers the exponent question. For a positive base other than one, the logarithm undoes exponentiation. Real logarithms require positive arguments. The inverse exchanges the original function domain and range.
Lesson 2.5
A logarithm gives an exponent. Product and quotient rules require positive arguments over the real numbers. Apply the rules to factors, not terms of a sum. The power rule moves an exponent outside a logarithm when the relevant real-domain conditions hold.
Lesson 2.6
Use inverse operations to solve exponential and logarithmic equations. A logarithmic equation over the reals requires every original logarithm argument to be positive. Solving a transformed equation can produce candidates outside that domain, so check the original expressions.
Lesson 2.7
Taking the logarithm of positive outputs turns an exponential model into a linear relationship. A straight pattern on a semi-log plot supports exponential growth or decay. The slope depends on the logarithm base.
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