Inside This Unit: The Full Breakdown
Exponential and logarithmic functions model multiplicative change. This unit connects arithmetic and geometric sequences to linear and exponential models, develops the properties of exponents and logarithms, and solves exponential and logarithmic equations.
Why it matters
Exponential and logarithmic reasoning is essential for modeling growth, decay, and many AP free-response contexts. Logarithms are the tool for solving for an unknown exponent, a skill tested in both multiple choice and free response.
Key concepts
- Arithmetic sequences add a common difference (linear); geometric sequences multiply by a common ratio (exponential).
- An exponential function f(x) = a·b^x has constant ratio over equal input intervals; b > 1 grows and 0 < b < 1 decays.
- Logarithms invert exponentials: log_b(y) = x means b^x = y.
- The product, quotient, and power properties of logarithms convert products to sums and exponents to coefficients.
Sequences and the Shape of Change
Arithmetic sequences change by a constant additive amount and model linear behavior; geometric sequences change by a constant multiplicative factor and model exponential behavior. Recognizing whether data has constant differences (linear) or constant ratios (exponential) is the first step in choosing a model.
Exponential Models, e, and Decay
An exponential model multiplies a starting value by a growth or decay factor repeatedly. A growth rate r gives factor (1 + r); decay gives a factor between 0 and 1. The natural base e arises from continuous growth. Half-life and doubling-time problems are written as a·(1/2)^(t/h) or a·2^(t/d), where h and d are the half-life and doubling time.
Logarithms and Solving Equations
A logarithm answers "what exponent?" Use the power property to bring a variable exponent down as a coefficient, then isolate it: solving b^x = k gives x = ln(k)/ln(b). For logarithmic equations, combine into a single logarithm, rewrite in exponential form, and always check for extraneous solutions, since logarithms require positive arguments.
AP exam tip
When solving an exponential equation, take the logarithm of both sides immediately and use the power property; for logarithmic equations, always verify each solution keeps every logarithm argument positive, discarding extraneous roots.
Connections to other units
- Unit 1: Exponential functions contrast with polynomial growth studied earlier.
- Unit 3: Sinusoidal models share the same amplitude/midline modeling logic as exponential models.
- Unit 4: Logarithmic scaling supports analyzing data linearized on semi-log plots.
