3.3 Exponential, Logarithmic, and Sequence Models
Key Takeaways
- 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.
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.
For equally spaced inputs n = 0, 1, 2, 3, the outputs are 5, 8, 12.8, 20.48. Which model best describes the pattern?
Solve log_3(x - 2) + log_3(x + 2) = 2 over the real numbers.
An arithmetic sequence has first term 7 and common difference -3. Which formula gives a_n?