Unit 2: Exponential and Logarithmic Functions

AP Precalculus: 47 practice questions with detailed explanations.

Unit Study Guide

Core idea

Exponentials model repeated multiplication; logarithms are their inverse - the exponent needed to reach a value.

Exponential functions

f(x) = a*bˣ with b > 0, b ≠ 1: growth if b > 1, decay if 0 < b < 1. Interpret b as PERCENT change: b = 1.07 means +7% per unit; b = 0.85 means -15%. The natural base e (~2.718) appears in continuous growth models. Before transformations, the horizontal asymptote sits at y = 0.

Logarithms

logₐ(c) answers: a raised to WHAT power gives c? Properties mirror exponents:

  • Product: log(mn) = log m + log n
  • Quotient: log(m/n) = log m - log n
  • Power: log(mᵖ) = p log m
  • Change of base: logₐ c = ln c / ln a

    • Solve exponential equations by matching bases or taking logs of both sides; ALWAYS check domains (log arguments must be positive).

    Data modeling

    Linear data grow by constant DIFFERENCES; exponential data grow by constant RATIOS. On a semi-log plot (log y vs x) exponential data fall on a straight line - that is the tell for choosing an exponential regression.

    Top 5 Concepts to Master

    1. 1Rewrite between exponential and logarithmic forms fluently.
    2. 2Condense and expand expressions with log properties.
    3. 3Solve exponential equations using matching bases or logs, checking domain.
    4. 4Interpret b in a*bˣ as percent growth or decay.
    5. 5Choose linear versus exponential models using ratios and semi-log plots.

    Key Terms & Definitions

    Practice with Flashcards
    Exponential function

    f(x) = a*bˣ with constant base b > 0, b ≠ 1.

    Growth/decay factor

    Base b read as percent change: 1.06 = +6% per step, 0.94 = -6% per step.

    Logarithm

    logₐ c is the exponent t satisfying aᵗ = c.

    Product property

    logₐ(mn) = logₐ m + logₐ n.

    Power property

    logₐ(mᵖ) = p*logₐ m.

    Change of base

    logₐ c = (ln c)/(ln a); lets any calculator evaluate any log.

    Semi-log plot

    Graph of log y against x; a straight line signals exponential data.

    Natural base e

    About 2.71828; the base for continuously compounding models.

    Common Misconceptions: Exam Traps

    log(m + n) = log m + log n.

    Correct: Logs turn MULTIPLICATION into addition: log(mn) = log m + log n. No rule exists for logs of sums.

    Adding 5 repeatedly models exponential growth.

    Correct: Repeated ADDITION is linear; exponential growth multiplies by a constant ratio.

    ln is a different kind of operation than log.

    Correct: ln is just log base e - same rules, special base.

    You can take a log of zero or negatives.

    Correct: Arguments must be positive; solving can create extraneous roots, so verify.

    Straight scatterplots imply exponential models.

    Correct: Raw-linear points suggest LINEAR models; exponential data straighten only on SEMI-log axes.

    Featured Practice Questions

    Question Bank Breakdown

    By difficulty

    easy 21medium 22hard 4

    By topic

    Logarithmic Functions 11Exponential Function Context and Data Modeling 9Logarithmic Function Manipulation 8Exponential Functions 8Logarithmic Function Context and Data Modeling 6Exponential Function Manipulation 3Semi-Log Plots 2

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