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3.3 Exponential, Logarithmic, and Sequence Models

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AEPA Mathematics NT304

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1 Test Framework, Processes & Number Sense

1.1 Current Test Facts and Framework 1.2 Mathematical Reasoning, Logic, and Proof 1.3 Real-Number Structure and Number Theory 1.4 Ratios, Rates, Estimation, and Units 1.5 Sets, Relations, and Representations

2 Algebra, Equations & Structures

2.1 Expressions, Equations, and Inequalities 2.2 Systems, Matrices, and Linear Relationships 2.3 Polynomial and Rational Expressions 2.4 Radical, Absolute-Value, and Piecewise Relationships 2.5 Algebraic Modeling and Error Analysis

3 Functions & Modeling

3.1 Function Notation, Domain, Range, and Composition 3.2 Linear, Quadratic, and Polynomial Functions 3.3 Exponential, Logarithmic, and Sequence Models 3.4 Inverse Functions and Transformations 3.5 Model Selection and Representation

4 Geometry, Measurement & Coordinate Math

4.1 Euclidean Geometry, Congruence, and Similarity 4.2 Measurement, Precision, Error, Area, and Volume 4.3 Circles, Polygons, and Solid Geometry 4.4 Coordinate Geometry, Vectors, and Conics 4.5 Transformations and Non-Euclidean Ideas

5 Trigonometry & Calculus

5.1 Unit Circle, Functions, and Graphs 5.2 Trigonometric Identities and Equations 5.3 Limits, Continuity, and Rates of Change 5.4 Derivatives and Applications 5.5 Integrals, Accumulation, and Area

6 Statistics, Probability & Discrete Math

6.1 Descriptive Statistics and Data Displays 6.2 Sampling, Bias, Correlation, and Inference 6.3 Probability, Counting, and Distributions 6.4 Sequences, Series, and Recursion 6.5 Graphs, Networks, and Discrete Structures

3.3 Exponential, Logarithmic, and Sequence Models

Key Takeaways Explain This Topic AI

Exponential models fit constant multiplicative change, not merely any increasing pattern.

Logarithms answer exponent questions and require positive arguments with valid bases greater than 0 and not equal to 1.

Arithmetic sequences are discrete linear models; geometric sequences are discrete exponential models.

Percent growth and decay are multiplication by 1 + r or 1 - r, so repeated percent change is not additive.

Last updated: June 2026

Multiplicative Change and Its Inverse

The AEPA/NES Mathematics profile explicitly includes exponential and logarithmic functions in the Patterns, Algebra, and Functions domain. These questions often look computational, but the central decision is conceptual: does the situation change by adding the same amount or by multiplying by the same factor?

An exponential function can be written as y = a b^x, where a is the initial value when x = 0 and b is the growth or decay factor. If b > 1, the model grows. If 0 < b < 1, the model decays. A 7% increase means multiply by 1.07 each period. A 7% decrease means multiply by 0.93 each period. The base is not the percent; it is the multiplier.

A logarithm is the inverse of an exponential expression. log_b(M) = p means b^p = M. The base b must be positive and not equal to 1, and the argument M must be positive. This restriction is a common source of distractors. If a candidate solves log_2(x - 5) = 3 and reports x = 8, the algebra mistake is treating the log as subtraction. The correct exponential form is x - 5 = 2^3, so x = 13.

Laws That Must Keep Their Conditions

  • Law Use Condition to remember
  • b^m b^n = b^(m+n) Same-base products Bases must match
  • (b^m)^n = b^(mn) Power of a power Applies to the exponent, not the base alone
  • log_b(MN) = log_b M + log_b N Products inside logs M and N must be positive
  • log_b(M/N) = log_b M - log_b N Quotients inside logs Denominator argument must be positive
  • p log_b M = log_b(M^p) Move coefficient to exponent Does not distribute over sums

These laws are reversible only when the expressions are defined. For example, log(x - 1) + log(x + 4) = log(10) becomes log((x - 1)(x + 4)) = log(10), but the final answer must satisfy x - 1 > 0 and x + 4 > 0. Solving the resulting quadratic without checking the original log arguments can include an extraneous value.

Worked Example: Growth with Time Units

A culture begins with 120 cells and doubles every 5 hours. The model is N(t) = 120(2)^(t/5), not 120(2)^t, because one exponent unit represents 5 hours. After 15 hours, t/5 = 3, so N(15) = 120(2^3) = 960. To find when the culture reaches 3840 cells, solve 3840 = 120(2)^(t/5). Divide by 120 to get 32 = 2^(t/5). Since 32 = 2^5, t/5 = 5 and t = 25 hours.

If the same problem used an 8% hourly increase, the model would be N(t) = 120(1.08)^t. To solve 500 = 120(1.08)^t, use logarithms: t = log(500/120)/log(1.08). The base of the logarithm may be common log or natural log as long as the same base is used in numerator and denominator.

Sequences as Discrete Models

A sequence is a function with inputs usually restricted to positive integers or nonnegative integers. Arithmetic sequences have constant difference and are discrete linear models: a_n = a_1 + (n - 1)d. Geometric sequences have constant ratio and are discrete exponential models: a_n = a_1 r^(n - 1). Recursive definitions give starting values and a rule using previous terms.

The discrete nature matters. A sequence can model the number of chairs in row n, but n = 2.5 has no contextual meaning. A continuous exponential function can model population over time, but the model may still be an approximation rather than exact counts at every instant.

Common Misconceptions to Diagnose

Teacher-certification questions may ask what a student misunderstood. If a student predicts 10%, then another 10%, means a 20% total increase, the misconception is additive treatment of multiplicative change. The actual factor is 1.1 times 1.1 = 1.21, a 21% increase. If a student changes log_b(M + N) into log_b M + log_b N, the misconception is inventing a false sum law. If a student identifies any upward curve as exponential, ask whether ratios are constant. Quadratic outputs can rise quickly too, but their second differences, not ratios, are constant.

For multiple-choice work, test a few values. Exponential growth eventually outpaces polynomial growth, but early values may not reveal that. Use the model's structure, domain, and units before relying on a calculator table.

Graph Checks and Parameter Meaning

Graph features give fast checks. Exponential graphs with positive initial value have a horizontal asymptote at y = 0 before vertical shifts; they do not cross that asymptote in the basic model. Logarithmic graphs have a vertical asymptote where the argument equals 0 and pass through a point that corresponds to exponent 0, since log_b(1) = 0. These landmarks help reject answer choices that have the right formula type but the wrong shift.

Parameter interpretation should use units. In P(t) = 450(1.03)^t, 450 is the initial amount and 1.03 is a growth factor per one t-unit. If t is months, the increase is 3% per month, not per year. In a geometric sequence a_n = 450(1.03)^(n - 1), the exponent n - 1 appears because the first term has not yet been multiplied by the ratio.